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Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
acute triangle
base angles
congruence transformation
kuhn·GROO·uhns
congruent triangles
coordinate proof
corollary
equiangular triangle
equilateral triangle
exterior angle
(continued on the next page)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
flow proof
included angle
included side
isosceles triangle
obtuse triangle
remote interior angles
right triangle
scalene triangle
SKAY·leen
vertex angle
⎧ ⎪ ⎨ ⎪ ⎩
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 4. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 4.1Angle Sum Theorem
Theorem 4.2Third Angle Theorem
Theorem 4.3Exterior Angle Theorem
Theorem 4.4
Theorem 4.5Angle-Angle-Side Congruence (AAS)
Theorem 4.6Leg-Leg Congruence (LL)
Theorem 4.7Hypotenuse-Angle Congruence (HA)
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 4.8Leg-Angle Congruence (LA)
Theorem 4.9Isosceles Triangle Theorem
Theorem 4.10
Postulate 4.1Side-Side-Side Congruence (SSS)
Postulate 4.2Side-Angle-Side Congruence (SAS)
Postulate 4.3Angle-Side-Angle Congruence (ASA)
Postulate 3.4Hypotenuse-Leg Congruence(HL)
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Study Guide and InterventionClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 183 Glencoe Geometry
Less
on
4-1
Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.
• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.
• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.
• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.
• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.
Classify each triangle.
a.
All three angles are congruent, so all three angles have measure 60°.The triangle is an equiangular triangle.
b.
The triangle has one angle that is obtuse. It is an obtuse triangle.
c.
The triangle has one right angle. It is a right triangle.
Classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
4. 5. 6.60!
28! 92!F D
B
45!
45!90!X Y
W
65! 65!
50!
U V
T
60! 60!
60!
Q
R S
120!
30! 30!N O
P
67!
90! 23!
K
L M
90!
60! 30!
G
H J
25!35!
120!
D F
E
60!
A
B C
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 184 Glencoe Geometry
Classify Triangles by Sides You can classify a triangle by the measures of its sides.Equal numbers of hash marks indicate congruent sides.
• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.
• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.
• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.
Classify each triangle.
a. b. c.
Two sides are congruent. All three sides are The triangle has no pairThe triangle is an congruent. The triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle.
Classify each triangle as equilateral, isosceles, or scalene.
1. 2. 3.
4. 5. 6.
7. Find the measure of each side of equilateral !RST with RS ! 2x " 2, ST ! 3x,and TR ! 5x # 4.
8. Find the measure of each side of isosceles !ABC with AB ! BC if AB ! 4y,BC ! 3y " 2, and AC ! 3y.
9. Find the measure of each side of !ABC with vertices A(#1, 5), B(6, 1), and C(2, #6).Classify the triangle.
D E
F
x
x x8x
32x
32x
B
CA
UW
S
12 17
19Q O
MG
K I18
18 182
1
3!"G C
A
23 12
15X V
TN
R PL J
H
Study Guide and Intervention (continued)
Classifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
ExampleExample
ExercisesExercises
Skills PracticeClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 185 Glencoe Geometry
Less
on
4-1
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
4. 5. 6.
Identify the indicated type of triangles.
7. right 8. isosceles
9. scalene 10. obtuse
ALGEBRA Find x and the measure of each side of the triangle.
11. !ABC is equilateral with AB! 3x # 2, BC ! 2x " 4, and CA ! x " 10.
12. !DEF is isosceles, "D is the vertex angle, DE ! x " 7, DF ! 3x # 1, and EF ! 2x " 5.
Find the measures of the sides of !RST and classify each triangle by its sides.
13. R(0, 2), S(2, 5), T(4, 2)
14. R(1, 3), S(4, 7), T(5, 4)
E CD
A B
© Glencoe/McGraw-Hill 186 Glencoe Geometry
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
Identify the indicated type of triangles if A!B! " A!D! " B!D! " D!C!, B!E! " E!D!, A!B! ⊥ B!C!, and E!D! ⊥ D!C!.
4. right 5. obtuse
6. scalene 7. isosceles
ALGEBRA Find x and the measure of each side of the triangle.
8. !FGH is equilateral with FG ! x " 5, GH ! 3x # 9, and FH ! 2x # 2.
9. !LMN is isosceles, "L is the vertex angle, LM ! 3x # 2, LN ! 2x " 1, and MN ! 5x # 2.
Find the measures of the sides of !KPL and classify each triangle by its sides.
10. K(#3, 2) P(2, 1), L(#2, #3)
11. K(5, #3), P(3, 4), L(#1, 1)
12. K(#2, #6), P(#4, 0), L(3, #1)
13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor.How many right angles are there?
A CD
EB
Practice Classifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
Reading to Learn MathematicsClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 187 Glencoe Geometry
Less
on
4-1
Pre-Activity Why are triangles important in construction?
Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.
• Why are triangles used for braces in construction rather than other shapes?
• Why do you think that isosceles triangles are used more often thanscalene triangles in construction?
Reading the Lesson1. Supply the correct numbers to complete each sentence.
a. In an obtuse triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
b. In an acute triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
c. In a right triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
2. Determine whether each statement is always, sometimes, or never true.a. A right triangle is scalene.b. An obtuse triangle is isosceles.c. An equilateral triangle is a right triangle.d. An equilateral triangle is isosceles.e. An acute triangle is isosceles.f. A scalene triangle is obtuse.
3. Describe each triangle by as many of the following words as apply: acute, obtuse, right,scalene, isosceles, or equilateral.a. b. c.
Helping You Remember4. A good way to remember a new mathematical term is to relate it to a nonmathematical
definition of the same word. How is the use of the word acute, when used to describeacute pain, related to the use of the word acute when used to describe an acute angle oran acute triangle?
5
34
135!80!
70!
30!
© Glencoe/McGraw-Hill 188 Glencoe Geometry
Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the texteasier to understand.
Consider three points, A, B, and C on a coordinate grid.The y-coordinates of A and B are the same. The x-coordinate of B isgreater than the x-coordinate of A. Both coordinates of C are greaterthan the corresponding coordinates of B. Is triangle ABC acute, right,or obtuse?
To answer this question, first draw a sample triangle that fits the description.
Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B.
From the diagram you can see that triangle ABCmust be obtuse.
Answer each question. Draw a simple triangle on the grid above to help you.
1. Consider three points, R, S, and 2. Consider three noncollinear points,T on a coordinate grid. The J, K, and L on a coordinate grid. Thex-coordinates of R and S are the y-coordinates of J and K are thesame. The y-coordinate of T is same. The x-coordinates of K and Lbetween the y-coordinates of R are the same. Is triangle JKL acute,and S. The x-coordinate of T is less right, or obtuse?than the x-coordinate of R. Is angleR of triangle RST acute, right, or obtuse?
3. Consider three noncollinear points, 4. Consider three points, G, H, and ID, E, and F on a coordinate grid. on a coordinate grid. Points G and The x-coordinates of D and E are H are on the positive y-axis, andopposites. The y-coordinates of D and the y-coordinate of G is twice the E are the same. The x-coordinate of y-coordinate of H. Point I is on the F is 0. What kind of triangle must positive x-axis, and the x-coordinate!DEF be: scalene, isosceles, or of I is greater than the y-coordinateequilateral? of G. Is triangle GHI scalene,
isosceles, or equilateral?
BA
Q
x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
ExampleExample
Study Guide and InterventionAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 189 Glencoe Geometry
Less
on
4-2
Angle Sum Theorem If the measures of two angles of a triangle are known,the measure of the third angle can always be found.
Angle Sum The sum of the measures of the angles of a triangle is 180.Theorem In the figure at the right, m"A " m"B " m"C ! 180.
CA
B
Find m"T.
m"R " m"S " m"T ! 180 Angle SumTheorem
25 " 35 " m"T ! 180 Substitution
60 " m"T ! 180 Add.
m"T ! 120 Subtract 60from each side.
35!
25!R T
S
Find the missing angle measures.
m"1 " m"A " m"B ! 180 Angle Sum Theorem
m"1 " 58 " 90 ! 180 Substitution
m"1 " 148 ! 180 Add.
m"1 ! 32 Subtract 148 fromeach side.
m"2 ! 32 Vertical angles arecongruent.
m"3 " m"2 " m"E ! 180 Angle Sum Theorem
m"3 " 32 " 108 ! 180 Substitution
m"3 " 140 ! 180 Add.
m"3 ! 40 Subtract 140 fromeach side.
58!
90!
108!
12 3
E
DAC
B
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the measure of each numbered angle.
1. 2.
3. 4.
5. 6. 20!
152!
DG
A
130!60!
1 2
S
R
T W
Q
O
NM
P58!
66!
50!
321
V
W T
U
30!
60!
2
1
S
Q R30!
1
90!
62!
1 N
M
P
© Glencoe/McGraw-Hill 190 Glencoe Geometry
Exterior Angle Theorem At each vertex of a triangle, the angle formed by one sideand an extension of the other side is called an exterior angle of the triangle. For eachexterior angle of a triangle, the remote interior angles are the interior angles that are notadjacent to that exterior angle. In the diagram below, "B and "A are the remote interiorangles for exterior "DCB.
Exterior AngleThe measure of an exterior angle of a triangle is equal to
Theoremthe sum of the measures of the two remote interior angles.m"1 ! m"A " m"B
AC
B
D1
Study Guide and Intervention (continued)
Angles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
Find m"1.
m"1 ! m"R " m"S Exterior Angle Theorem
! 60 " 80 Substitution
! 140 Add.
R T
S
60!
80!
1
Find x.
m"PQS ! m"R " m"S Exterior Angle Theorem
78 ! 55 " x Substitution
23 ! x Subtract 55 from each side.
S R
Q
P
55!
78!
x!
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the measure of each numbered angle.
1. 2.
3. 4.
Find x.
5. 6. E
FGH
58!
x !
x !B
A
DC
95!
2x ! 145!
U T
SRV
35! 36!
80!
13
2
POQ
N M60!
60!3 2
1
B C D
A
25!
35!
12Y Z W
X
65!
50!
1
Skills PracticeAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 191 Glencoe Geometry
Less
on
4-2
Find the missing angle measures.
1. 2.
Find the measure of each angle.
3. m"1
4. m"2
5. m"3
Find the measure of each angle.
6. m"1
7. m"2
8. m"3
Find the measure of each angle.
9. m"1
10. m"2
11. m"3
12. m"4
13. m"5
Find the measure of each angle.
14. m"1
15. m"2 63!
1
2D
A C
B
80!
60!
40!
105!
1 4 52
3
150!55!
70!
1 2
3
85! 55!
40!
1 2
3
146!
TIGERS80!
73!
© Glencoe/McGraw-Hill 192 Glencoe Geometry
Find the missing angle measures.
1. 2.
Find the measure of each angle.
3. m"1
4. m"2
5. m"3
Find the measure of each angle.
6. m"1
7. m"4
8. m"3
9. m"2
10. m"5
11. m"6
Find the measure of each angle if "BAD and "BDC are right angles and m"ABC " 84.
12. m"1
13. m"2
14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m"1.
145!1
64!1
2A
BC
D
118!36!
68!
70!
65!
82!
1
2
3 4
5
6
58!
39!
35!
12
3
40! 55!
72!
?
Practice Angles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
Reading to Learn MathematicsAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 193 Glencoe Geometry
Less
on
4-2
Pre-Activity How are the angles of triangles used to make kites?
Read the introduction to Lesson 4-2 at the top of page 185 in your textbook.
The frame of the simplest kind of kite divides the kite into four triangles.Describe these four triangles and how they are related to each other.
Reading the Lesson
1. Refer to the figure.
a. Name the three interior angles of the triangle. (Use threeletters to name each angle.)
b. Name three exterior angles of the triangle. (Use three lettersto name each angle.)
c. Name the remote interior angles of "EAB.
d. Find the measure of each angle without using a protractor.
i. "DBC ii. "ABC iii. "ACF iv. "EAB
2. Indicate whether each statement is true or false. If the statement is false, replace theunderlined word or number with a word or number that will make the statement true.
a. The acute angles of a right triangle are .
b. The sum of the measures of the angles of any triangle is .
c. A triangle can have at most one right angle or angle.
d. If two angles of one triangle are congruent to two angles of another triangle, then thethird angles of the triangles are .
e. The measure of an exterior angle of a triangle is equal to the of themeasures of the two remote interior angles.
f. If the measures of two angles of a triangle are 62 and 93, then the measure of thethird angle is .
g. An angle of a triangle forms a linear pair with an interior angle of thetriangle.
Helping You Remember
3. Many students remember mathematical ideas and facts more easily if they see themdemonstrated visually rather than having them stated in words. Describe a visual wayto demonstrate the Angle Sum Theorem.
exterior
35
difference
congruent
acute
100
supplementary
39!
23!
EA B D
CF
© Glencoe/McGraw-Hill 194 Glencoe Geometry
Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.
In triangle ABC, m"A, is twice m"B, and m"Cis 8 more than m"B. What is the measure of each angle?
Write and solve an equation. Let x ! m"B.m"A " m"B " m"C ! 180
2x " x " (x " 8) ! 1804x " 8 ! 180
4x ! 172x ! 43
So, m"A ! 2(43) or 86, m"B ! 43, and m"C ! 43 " 8 or 51.
Solve each problem.
1. In triangle DEF, m"E is three times 2. In triangle RST, m"T is 5 more than m"D, and m"F is 9 less than m"E. m"R, and m"S is 10 less than m"T.What is the measure of each angle? What is the measure of each angle?
3. In triangle JKL, m"K is four times 4. In triangle XYZ, m"Z is 2 more than twicem"J, and m"L is five times m"J. m"X, and m"Y is 7 less than twice m"X.What is the measure of each angle? What is the measure of each angle?
5. In triangle GHI, m"H is 20 more than 6. In triangle MNO, m"M is equal to m"N,m"G, and m"G is 8 more than m"I. and m"O is 5 more than three times What is the measure of each angle? m"N. What is the measure of each angle?
7. In triangle STU, m"U is half m"T, 8. In triangle PQR, m"P is equal to and m"S is 30 more than m"T. What m"Q, and m"R is 24 less than m"P.is the measure of each angle? What is the measure of each angle?
9. Write your own problems about measures of triangles.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
ExampleExample
Study Guide and InterventionCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 195 Glencoe Geometry
Less
on
4-3
Corresponding Parts of Congruent TrianglesTriangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, !ABC # !RST.
If !XYZ " !RST, name the pairs of congruent angles and congruent sides."X # "R, "Y # "S, "Z # "TX$Y$ # R$S$, X$Z$ # R$T$, Y$Z$ # S$T$
Identify the congruent triangles in each figure.
1. 2. 3.
Name the corresponding congruent angles and sides for the congruent triangles.
4. 5. 6. R
T
U S
B D
CA
F G L K
JE
K
J
L
MC
D
A
B
CA
B
LJ
K
Y
X ZT
SR
AC
BR
T
S
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 196 Glencoe Geometry
Identify Congruence Transformations If two triangles are congruent, you canslide, flip, or turn one of the triangles and they will still be congruent. These are calledcongruence transformations because they do not change the size or shape of the figure.It is common to use prime symbols to distinguish between an original !ABC and atransformed !A$B$C$.
Name the congruence transformation that produces !A#B#C# from !ABC.The congruence transformation is a slide."A # "A$; "B # "B$; "C #"C$;A$B$ # A$$$B$$$; A$C$ # A$$$C$$$; B$C$ # B$$$C$$$
Describe the congruence transformation between the two triangles as a slide, aflip, or a turn. Then name the congruent triangles.
1. 2.
3. 4.
5. 6.
x
y
OM
N
P
N$
P$x
y
OA$ B$
C$
A B
C
x
y
OB$B
A
Cx
y
O
Q$ P$
P
Q
x
y
ON$
M$P$
N
MP
x
y
OR
S$T$
S
T
x
y
O
A$
B$B
C$A C
Study Guide and Intervention (continued)
Congruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
ExampleExample
ExercisesExercises
Skills PracticeCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 197 Glencoe Geometry
Less
on
4-3
Identify the congruent triangles in each figure.
1. 2.
3. 4.
Name the congruent angles and sides for each pair of congruent triangles.
5. !ABC # !FGH
6. !PQR # !STU
Verify that each of the following transformations preserves congruence, and namethe congruence transformation.
7. !ABC # !A$B$C$ 8. !DEF # !D$E$F $
x
y
OD$
E$
F$D
E
F
x
y
OA$
B$
C$
A
B
C
D
E
G
FRP
Q
S
WY
X C
AB
L
P
J
S
V
T
© Glencoe/McGraw-Hill 198 Glencoe Geometry
Identify the congruent triangles in each figure.
1. 2.
Name the congruent angles and sides for each pair of congruent triangles.
3. !GKP # !LMN
4. !ANC # !RBV
Verify that each of the following transformations preserves congruence, and namethe congruence transformation.
5. !PST # !P$S$T$ 6. !LMN # !L$M$N$
QUILTING For Exercises 7 and 8, refer to the quilt design.
7. Indicate the triangles that appear to be congruent.
8. Name the congruent angles and congruent sides of a pair of congruent triangles.
B
A
I
E
FH G
C D
x
y
O
M$
N$L$
M
NL
x
y
O
S$
T$P$
S
TP
MN
L
P
QD
R
SCA
B
Practice Congruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Reading to Learn MathematicsCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 199 Glencoe Geometry
Less
on
4-3
Pre-Activity Why are triangles used in bridges?
Read the introduction to Lesson 4-3 at the top of page 192 in your textbook.
In the bridge shown in the photograph in your textbook, diagonal braceswere used to divide squares into two isosceles right triangles. Why do youthink these braces are used on the bridge?
Reading the Lesson1. If !RST # !UWV, complete each pair of congruent parts.
"R # # "W "T #
R$T$ # # U$W$ # W$V$
2. Identify the congruent triangles in each diagram.a. b.
c. d.
3. Determine whether each statement says that congruence of triangles is reflexive,symmetric, or transitive.a. If the first of two triangles is congruent to the second triangle, then the second
triangle is congruent to the first.b. If there are three triangles for which the first is congruent to the second and the second
is congruent to the third, then the first triangle is congruent to the third.c. Every triangle is congruent to itself.
Helping You Remember4. A good way to remember something is to explain it to someone else. Your classmate Ben is
having trouble writing congruence statements for triangles because he thinks he has tomatch up three pairs of sides and three pairs of angles. How can you help him understandhow to write correct congruence statements more easily?
R T
US
V
N O P
QM
S
P R
Q
CA
B
D
© Glencoe/McGraw-Hill 200 Glencoe Geometry
Transformations in The Coordinate PlaneThe following statement tells one way to map preimage points to image points in the coordinate plane.
(x, y) → (x " 6, y # 3)
This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x " 6, y # 3).”With this transformation, for example, (3, 5) is mapped to (3 " 6, 5 # 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.
1. Does the transformation above appear to be a congruence transformation? Explain youranswer.
Draw the transformation image for each figure. Then tell whether thetransformation is or is not a congruence transformation.
2. (x, y) → (x # 4, y) 3. (x, y) → (x " 8, y " 7)
4. (x, y) → (#x , #y) 5. (x, y) → %#%12%x, y&
x
y
Ox
y
O
x
y
Ox
y
O
x
y
B
A
C X
Z
Y
O
(x, y ) → (x $ 6, y % 3)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Study Guide and InterventionProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 201 Glencoe Geometry
Less
on
4-4
SSS Postulate You know that two triangles are congruent if corresponding sides arecongruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate letsyou show that two triangles are congruent if you know only that the sides of one triangleare congruent to the sides of the second triangle.
SSS PostulateIf the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
Write a two-column proof.Given: A$B$ # D$B$ and C is the midpoint of A$D$.Prove: !ABC # !DBC
Statements Reasons
1. A$B$ # D$B$ 1. Given2. C is the midpoint of A$D$. 2. Given3. A$C$ # D$C$ 3. Definition of midpoint4. B$C$ # B$C$ 4. Reflexive Property of #5. !ABC # !DBC 5. SSS Postulate
Write a two-column proof.
B
CDA
ExampleExample
ExercisesExercises
1.
Given: A$B$ # X$Y$, A$C$ # X$Z$, B$C$ # Y$Z$Prove: !ABC # !XYZ
Statements Reasons
1.A!B! " X!Y! 1.Given
2.!ABC " !XYZ 2. SSS Post.
2.
Given: R$S$ # U$T$, R$T$ # U$S$Prove: !RST # !UTS
Statements Reasons
1.R!S! " U!T! 1. Given
2.S!T! " T!S! 2. Refl. Prop.3.!RST " !UTS 3. SSS Post.
T U
R S
B Y
CA XZ
© Glencoe/McGraw-Hill 202 Glencoe Geometry
SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.
SAS PostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
For each diagram, determine which pairs of triangles can beproved congruent by the SAS Postulate.
a. b. c.
In !ABC, the angle is not The right angles are The included angles, "1 “included” by the sides A$B$ congruent and they are the and "2, are congruent and A$C$. So the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the SAS Postulate. !DEF # !JGH by the for two parallel lines.
SAS Postulate. !PSR # !RQP by the SAS Postulate.
For each figure, determine which pairs of triangles can be proved congruent bythe SAS Postulate.
1. 2. 3.
4. 5. 6.
J H
GF
K
C
BA
D
V
T
W
M
P L
N
M
X
W Z
YQT
P
UN M
R
P Q
S
1
2 R
D H
F E
G
J
A
B C
X
Y Z
Study Guide and Intervention (continued)
Proving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
ExampleExample
ExercisesExercises
Skills PracticeProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 203 Glencoe Geometry
Less
on
4-4
Determine whether !ABC " !KLM given the coordinates of the vertices. Explain.
1. A(#3, 3), B(#1, 3), C(#3, 1), K(1, 4), L(3, 4), M(1, 6)
2. A(#4, #2), B(#4, 1), C(#1, #1), K(0, #2), L(0, 1), M(4, 1)
3. Write a flow proof.Given: P$R$ # D$E$, P$T$ # D$F$
"R # "E, "T # "FProve: !PRT # !DEF
Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.
4. 5. 6.
PR " DEGiven
PT " DF Given
"R " "E Given
"P " "D Third AngleTheorem
!PRT " !DEF SAS
"T " "F Given
T
R
P
F
E
D
© Glencoe/McGraw-Hill 204 Glencoe Geometry
Determine whether !DEF " !PQR given the coordinates of the vertices. Explain.
1. D(#6, 1), E(1, 2), F(#1, #4), P(0, 5), Q(7, 6), R(5, 0)
2. D(#7, #3), E(#4, #1), F(#2, #5), P(2, #2), Q(5, #4), R(0, #5)
3. Write a flow proof.Given: R$S$ # T$S$
V is the midpoint of R$T$.Prove: !RSV # !TSV
Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.
4. 5. 6.
7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in thediagram. How does he know that the lengths A$B$ and AB are equal?
A$B$
A B
C
RS " TSGiven
SV " SVReflexiveProperty
RV " VTDefinitionof midpoint
V is themidpoint of RT. Given
!RSV " !TSV SSS
S
R
V
T
Practice Proving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Reading to Learn MathematicsProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 205 Glencoe Geometry
Less
on
4-4
Pre-Activity How do land surveyors use congruent triangles?
Read the introduction to Lesson 4-4 at the top of page 200 in your textbook.
Why do you think that land surveyors would use congruent right trianglesrather than other congruent triangles to establish property boundaries?
Reading the Lesson
1. Refer to the figure.
a. Name the sides of !LMN for which "L is the included angle.
b. Name the sides of !LMN for which "N is the included angle.
c. Name the sides of !LMN for which "M is the included angle.
2. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate that you would use. If not, write not possible.a. b.
c. E$H$ and D$G$ bisect each other. d.
Helping You Remember
3. Find three words that explain what it means to say that two triangles are congruent andthat can help you recall the meaning of the SSS Postulate.
GE
F
HD
R
T
SU
G
FD
E
C
A
DB
L
N
M
© Glencoe/McGraw-Hill 206 Glencoe Geometry
Congruent Parts of Regular Polygonal RegionsCongruent figures are figures that have exactly the same size and shape. There are manyways to divide regular polygonal regions into congruent parts. Three ways to divide anequilateral triangular region are shown. You can verify that the parts are congruent bytracing one part, then rotating, sliding, or reflecting that part on top of the other parts.
1. Divide each square into four congruent parts. Use three different ways.
2. Divide each pentagon into five congruent parts. Use three different ways.
3. Divide each hexagon into six congruent parts. Use three different ways.
4. What hints might you give another student who is trying to divide figures like those into congruent parts?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Study Guide and InterventionProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 207 Glencoe Geometry
Less
on
4-5
ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two trianglesare congruent.
ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Find the missing congruent parts so that the triangles can beproved congruent by the ASA Postulate. Then write the triangle congruence.
a.
Two pairs of corresponding angles are congruent, "A # "D and "C # "F. If theincluded sides A$C$ and D$F$ are congruent, then !ABC # !DEF by the ASA Postulate.
b.
"R # "Y and S$R$ # X$Y$. If "S # "X, then !RST# !YXW by the ASA Postulate.
What corresponding parts must be congruent in order to prove that the trianglesare congruent by the ASA Postulate? Write the triangle congruence statement.
1. 2. 3.
4. 5. 6.
A C
B
E
D
S
V
UR T
D
A B
C
D CEA
B
YW
X
ZE A
BD
C
R T W Y
S X
A C
B
D F
E
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 208 Glencoe Geometry
AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.
AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding twoangles and side of a second triangle, then the two triangles are congruent.
You now have five ways to show that two triangles are congruent.• definition of triangle congruence • ASA Postulate• SSS Postulate • AAS Theorem• SAS Postulate
In the diagram, "BCA " "DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Postulate?A$C$ # A$C$ by the Reflexive Property of congruence. The congruent angles cannot be "1 and "2, because A$C$ would be the included side.If "B # "D, then !ABC # !ADC by the AAS Theorem.
In Exercises 1 and 2, draw and label !ABC and !DEF. Indicate which additionalpair of corresponding parts needs to be congruent for the triangles to becongruent by the AAS Theorem.
1. "A # "D; "B # "E 2. BC # EF; "A # "D
3. Write a flow proof.Given: "S # "U; T$R$ bisects "STU.Prove: "SRT # "URT
Given
Given
RT " RT Refl. Prop. of "
Def.of " bisectorTR bisects "STU.
!SRT " !URT
"STR " "UTR
AAS"SRT " "URTCPCTC
"S " "U
S
R T
U
BA
C
ED
F
CA
B
FD
E
D
C12A
B
Study Guide and Intervention (continued)
Proving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
ExampleExample
ExercisesExercises
Skills PracticeProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 209 Glencoe Geometry
Less
on
4-5
Write a flow proof.
1. Given: "N # "LJ$K$ # M$K$
Prove: !JKN # !MKL
2. Given: A$B$ # C$B$"A # "CD$B$ bisects "ABC.
Prove: A$D$ # C$D$
3. Write a paragraph proof.
Given: D$E$ || F$G$"E # "G
Prove: !DFG # !FDE
FG
D E
"A " "C
GivenAB " CB
Given CPCTCAD " CD
DB bisects "ABC. Given
!ABD " !CBDASA
"ABD " "CBDDef. of " bisector
A C
B
D
"N " "LGiven
JK " MK Given
"JKN " "MKL Vertical # are ".
!JKN " !MKLAAS
N
J
M
K L
© Glencoe/McGraw-Hill 210 Glencoe Geometry
1. Write a flow proof.Given: S is the midpoint of Q$T$.
Q$R$ || T$U$Prove: !QSR # !TSU
2. Write a paragraph proof.
Given: "D # "FG$E$ bisects "DEF.
Prove: D$G$ # F$G$
ARCHITECTURE For Exercises 3 and 4, use the following information.An architect used the window design in the diagram when remodeling an art studio. A$B$ and C$B$ each measure 3 feet.
3. Suppose D is the midpoint of A$C$. Determine whether !ABD # !CBD.Justify your answer.
4. Suppose "A # "C. Determine whether !ABD # !CBD. Justify your answer.
D
B
A C
D
G
F
E
"Q " "T
Given
QR || TU Given
Def.of midpoint
Alt. Int. # are ".
QS " TS S is the midpoint of QT.
!QSR " !TSUASA
"QSR " "TSUVertical # are ".
UQ S
RT
Practice Proving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Reading to Learn MathematicsProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 211 Glencoe Geometry
Less
on
4-5
Pre-Activity How are congruent triangles used in construction?Read the introduction to Lesson 4-5 at the top of page 207 in your textbook.Which of the triangles in the photograph in your textbook appear to becongruent?
Reading the Lesson1. Explain in your own words the difference between how the ASA Postulate and the AAS
Theorem are used to prove that two triangles are congruent.
2. Which of the following conditions are sufficient to prove that two triangles are congruent?A. Two sides of one triangle are congruent to two sides of the other triangle.B. The three sides of one triangles are congruent to the three sides of the other triangle.C. The three angles of one triangle are congruent to the three angles of the other triangle.D. All six corresponding parts of two triangles are congruent.E. Two angles and the included side of one triangle are congruent to two sides and the
included angle of the other triangle.F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a
nonincluded angle of the other triangle.G. Two angles and a nonincluded side of one triangle are congruent to two angles and
the corresponding nonincluded side of the other triangle.H. Two sides and the included angle of one triangle are congruent to two sides and the
included angle of the other triangle.I. Two angles and a nonincluded side of one triangle are congruent to two angles and a
nonincluded side of the other triangle.
3. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate or theorem that you would use. If not, write not possible.a. b. T is the midpoint of R$U$.
Helping You Remember4. A good way to remember mathematical ideas is to summarize them in a general statement.
If you want to prove triangles congruent by using three pairs of corresponding parts,what is a good way to remember which combinations of parts will work?
R
S
T
U
V
A DCB
E
© Glencoe/McGraw-Hill 212 Glencoe Geometry
Congruent Triangles in the Coordinate PlaneIf you know the coordinates of the vertices of two triangles in the coordinateplane, you can often decide whether the two triangles are congruent. Theremay be more than one way to do this.
1. Consider ! ABD and !CDB whose vertices have coordinates A(0, 0),B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what youknow about congruent triangles and the coordinate plane to show that ! ABD # !CDB. You may wish to make a sketch to help get you started.
2. Consider !PQR and !KLM whose vertices are the following points.
P(1, 2) Q(3, 6) R(6, 5)K(#2, 1) L(#6, 3) M(#5, 6)
Briefly describe how you can show that !PQR # !KLM.
3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Study Guide and InterventionIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 213 Glencoe Geometry
Less
on
4-6
Properties of Isosceles Triangles An isosceles triangle has two congruent sides.The angle formed by these sides is called the vertex angle. The other two angles are calledbase angles. You can prove a theorem and its converse about isosceles triangles.
• If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem)
• If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If A$B$ # C$B$, then "A # "C.
If "A # "C, then A$B$ # C$B$.
A
BC
Find x.
BC ! BA, so m"A ! m"C. Isos. Triangle Theorem
5x # 10 ! 4x " 5 Substitution
x # 10 ! 5 Subtract 4x from each side.
x ! 15 Add 10 to each side.
B
A
C (4x $ 5)!
(5x % 10)!
Find x.
m"S ! m"T, soSR ! TR. Converse of Isos. ! Thm.
3x # 13 ! 2x Substitution
3x ! 2x " 13 Add 13 to each side.
x ! 13 Subtract 2x from each side.
R T
S
3x % 13
2x
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.
7. Write a two-column proof.Given: "1 # "2Prove: A$B$ # C$B$
Statements Reasons
B
A C D
E
1 32
R S
T3x!
x!D
BG
L3x!
30!D
T Q
PK
(6x $ 6)! 2x!
W
Y Z3x!
S
V
T 3x % 6
2x $ 6R
P
Q2x !
40!
© Glencoe/McGraw-Hill 214 Glencoe Geometry
Properties of Equilateral Triangles An equilateral triangle has three congruentsides. The Isosceles Triangle Theorem can be used to prove two properties of equilateraltriangles.
1. A triangle is equilateral if and only if it is equiangular.2. Each angle of an equilateral triangle measures 60°.
Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle.Proof:Statements Reasons
1. !ABC is equilateral; P$Q$ || B$C$. 1. Given2. m"A ! m"B ! m"C ! 60 2. Each " of an equilateral ! measures 60°.3. "1 # "B, "2 # "C 3. If || lines, then corres. "s are #.4. m"1 ! 60, m"2 ! 60 4. Substitution5. !APQ is equilateral. 5. If a ! is equiangular, then it is equilateral.
Find x.
1. 2. 3.
4. 5. 6.
7. Write a two-column proof.Given: !ABC is equilateral; "1 # "2.Prove: "ADB # "CDB
Proof:
Statements Reasons
A
D
C
B12
R
O
HM 60!4x!
X
Z Y4x % 4
3x $ 8 60!
P Q
LV R60!
4x 40
L
N
M
K
!KLM is equilateral.
3x!G
J H
6x % 5 5x
D
F E6x!
A
B
P Q
C
1 2
Study Guide and Intervention (continued)
Isosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
ExampleExample
ExercisesExercises
Skills PracticeIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 215 Glencoe Geometry
Less
on
4-6
Refer to the figure.
1. If A$C$ # A$D$, name two congruent angles.
2. If B$E$ # B$C$, name two congruent angles.
3. If "EBA # "EAB, name two congruent segments.
4. If "CED # "CDE, name two congruent segments.
!ABF is isosceles, !CDF is equilateral, and m"AFD " 150.Find each measure.
5. m"CFD 6. m"AFB
7. m"ABF 8. m"A
In the figure, P!L! " R!L! and L!R! " B!R!.
9. If m"RLP ! 100, find m"BRL.
10. If m"LPR ! 34, find m"B.
11. Write a two-column proof.
Given: C$D$ # C$G$D$E$ # G$F$
Prove: C$E$ # C$F$
DE
FG
C
R P
BL
D
C
F
B
35!
A E
D
C
B
A E
© Glencoe/McGraw-Hill 216 Glencoe Geometry
Refer to the figure.
1. If R$V$ # R$T$, name two congruent angles.
2. If R$S$ # S$V$, name two congruent angles.
3. If "SRT # "STR, name two congruent segments.
4. If "STV # "SVT, name two congruent segments.
Triangles GHM and HJM are isosceles, with G!H! " M!H!and H!J! " M!J!. Triangle KLM is equilateral, and m"HMK " 50.Find each measure.
5. m"KML 6. m"HMG 7. m"GHM
8. If m"HJM ! 145, find m"MHJ.
9. If m"G ! 67, find m"GHM.
10. Write a two-column proof.
Given: D$E$ || B$C$"1 # "2
Prove: A$B$ # A$C$
11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle. Lincoln Hawks
E
D B
C
A1
23
4
G
M
LK
J
H
U
R
TV
S
Practice Isosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
Reading to Learn MathematicsIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 217 Glencoe Geometry
Less
on
4-6
Pre-Activity How are triangles used in art?
Read the introduction to Lesson 4-6 at the top of page 216 in your textbook.
• Why do you think that isosceles and equilateral triangles are used moreoften than scalene triangles in art?
• Why might isosceles right triangles be used in art?
Reading the Lesson1. Refer to the figure.
a. What kind of triangle is !QRS?
b. Name the legs of !QRS.
c. Name the base of !QRS.
d. Name the vertex angle of !QRS.
e. Name the base angles of !QRS.
2. Determine whether each statement is always, sometimes, or never true.
a. If a triangle has three congruent sides, then it has three congruent angles.
b. If a triangle is isosceles, then it is equilateral.
c. If a right triangle is isosceles, then it is equilateral.
d. The largest angle of an isosceles triangle is obtuse.
e. If a right triangle has a 45° angle, then it is isosceles.
f. If an isosceles triangle has three acute angles, then it is equilateral.
g. The vertex angle of an isosceles triangle is the largest angle of the triangle.
3. Give the measures of the three angles of each triangle.
a. an equilateral triangle
b. an isosceles right triangle
c. an isosceles triangle in which the measure of the vertex angle is 70
d. an isosceles triangle in which the measure of a base angle is 70
e. an isosceles triangle in which the measure of the vertex angle is twice the measure ofone of the base angles
Helping You Remember4. If a theorem and its converse are both true, you can often remember them most easily by
combining them into an “if-and-only-if” statement. Write such a statement for the IsoscelesTriangle Theorem and its converse.
R
Q
S
© Glencoe/McGraw-Hill 218 Glencoe Geometry
Triangle ChallengesSome problems include diagrams. If you are not sure how to solve theproblem, begin by using the given information. Find the measures of as manyangles as you can, writing each measure on the diagram. This may give youmore clues to the solution.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
1. Given: BE ! BF, "BFG # "BEF #"BED, m"BFE ! 82 andABFG and BCDE each haveopposite sides parallel andcongruent.
Find m"ABC.
3. Given: m"UZY ! 90, m"ZWX ! 45,!YZU # !VWX, UVXY is asquare (all sides congruent, allangles right angles).
Find m"WZY.
2. Given: AC ! AD, and A$B$#B$D$,m"DAC ! 44 andC$E$ bisects "ACD.
Find m"DEC.
4. Given: m"N ! 120, J$N$ # M$N$,!JNM # !KLM.
Find m"JKM.J
K
L
MN
A
D C
BE
U VW
XYZ
A
G DF E
CB
Study Guide and InterventionTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 219 Glencoe Geometry
Less
on
4-7
Position and Label Triangles A coordinate proof uses points, distances, and slopes toprove geometric properties. The first step in writing a coordinate proof is to place a figure onthe coordinate plane and label the vertices. Use the following guidelines.
1. Use the origin as a vertex or center of the figure.2. Place at least one side of the polygon on an axis.3. Keep the figure in the first quadrant if possible.4. Use coordinates that make the computations as simple as possible.
Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis.Start with R(0, 0). If RT is a, then another vertex is T(a, 0).
For vertex S, the x-coordinate is %a2%. Use b for the y-coordinate,
so the vertex is S%%a2%, b&.
Find the missing coordinates of each triangle.
1. 2. 3.
Position and label each triangle on the coordinate plane.
4. isosceles triangle 5. isosceles right !DEF 6. equilateral triangle !EQI!RST with base R$S$ with legs e units long with vertex Q(0, a) and4a units long sides 2b units long
x
y
I(b, 0)E(–b, 0)
Q(0, a)
x
y
E(e, 0)
F(e, e)
D(0, 0)x
y T(2a, b)
R(0, 0) S(4a, 0)
x
y
G(2g, 0)
F(?, b)
E(?, ?)x
y
S(2a, 0)
T(?, ?)
R(0, 0)x
y
B(2p, 0)
C(?, q)
A(0, 0)
x
y
T(a, 0)R(0, 0)
S#a–2, b$
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 220 Glencoe Geometry
Write Coordinate Proofs Coordinate proofs can be used to prove theorems and toverify properties. Many coordinate proofs use the Distance Formula, Slope Formula, orMidpoint Theorem.
Prove that a segment from the vertex angle of an isosceles triangle to the midpoint of the base is perpendicular to the base.First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(#a, 0), and S(0, c). Then U(0, 0) is the midpoint of R$T$.
Given: Isosceles !RST; U is the midpoint of base R$T$.Prove: S$U$ ⊥ R$T$
Proof:U is the midpoint of R$T$ so the coordinates of U are %%#a
2" a%, %
0 "2
0%& ! (0, 0). Thus S$U$ lies on
the y-axis, and !RST was placed so R$T$ lies on the x-axis. The axes are perpendicular, so S$U$ ⊥ R$T$.
Prove that the segments joining the midpoints of the sides of a right triangle forma right triangle.
C(2a, 0)
B(0, 2b)
P
Q
R
A(0, 0)
x
y
T(a, 0)U(0, 0)R(–a, 0)
S(0, c)
Study Guide and Intervention (continued)
Triangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
ExampleExample
ExercisesExercises
Skills PracticeTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 221 Glencoe Geometry
Less
on
4-7
Position and label each triangle on the coordinate plane.
1. right !FGH with legs 2. isosceles !KLP with 3. isosceles !AND witha units and b units base K$P$ 6b units long base A$D$ 5a long
Find the missing coordinates of each triangle.
4. 5. 6.
7. 8. 9.
10. Write a coordinate proof to prove that in an isosceles right triangle, the segment fromthe vertex of the right angle to the midpoint of the hypotenuse is perpendicular to thehypotenuse.
Given: isosceles right !ABC with "ABC the right angle and M the midpoint of A$C$Prove: B$M$ ⊥ A$C$
C(2a, 0)
A(0, 2a)
M
B(0, 0)
x
y
U(a, 0)
T(?, ?)
S(–a, 0)x
y
P(7b, 0)
R(?, ?)
N(0, 0)x
y
Q(?, ?)
R(2a, b)
P(0, 0)
x
y
N(3b, 0)
M(?, ?)
O(0, 0)x
y
Y(2b, 0)
Z(?, ?)
X(0, 0)x
y
B(2a, 0)
A(0, ?)
C(0, 0)
x
yN #5–
2a, b$
A(0, 0) D(5a, 0)x
y L(3b, c)
K(0, 0) P(6b, 0)x
yF(0, a)
G(0, 0) H(b, 0)
© Glencoe/McGraw-Hill 222 Glencoe Geometry
Position and label each triangle on the coordinate plane.
1. equilateral !SWY with 2. isosceles !BLP with 3. isosceles right !DGJsides %
14%a long base B$L$ 3b units long with hypotenuse D$J$ and
legs 2a units long
Find the missing coordinates of each triangle.
4. 5. 6.
NEIGHBORHOODS For Exercises 7 and 8, use the following information.Karina lives 6 miles east and 4 miles north of her high school. After school she works parttime at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.
7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall areat the vertices of a right triangle.
Given: !SKMProve: !SKM is a right triangle.
8. Find the distance between the mall and Karina’s home.
x
y
S(0, 0)
K(6, 4)
M(–2, 3)
x
y
P(2b, 0)
M(0, ?)
N(?, 0)x
y
C(?, 0)
E(0, ?)
B(–3a, 0)x
y S(?, ?)
J(0, 0) R#1–3b, 0$
x
yD(0, 2a)
G(0, 0) J(2a, 0)x
yP #3–
2b, c$
B(0, 0) L(3b, 0)x
y Y #1–8a, b$
W #1–4a, 0$S(0, 0)
Practice Triangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
Reading to Learn MathematicsTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 223 Glencoe Geometry
Less
on
4-7
Pre-Activity How can the coordinate plane be useful in proofs?
Read the introduction to Lesson 4-7 at the top of page 222 in your textbook.
From the coordinates of A, B, and C in the drawing in your textbook, whatdo you know about !ABC?
Reading the Lesson
1. Find the missing coordinates of each triangle.a. b.
2. Refer to the figure.
a. Find the slope of S$R$ and the slope of S$T$.
b. Find the product of the slopes of S$R$ and S$T$. What does this tell you about S$R$ and S$T$?
c. What does your answer from part b tell you about !RST ?
d. Find SR and ST. What does this tell you about S$R$ and S$T$?
e. What does your answer from part d tell you about !RST?
f. Combine your answers from parts c and e to describe !RST as completely as possible.
g. Find m"SRT and m"STR.
h. Find m"OSR and m"OST.
Helping You Remember
3. Many students find it easier to remember mathematical formulas if they can put theminto words in a compact way. How can you use this approach to remember the slope andmidpoint formulas easily?
x
y
S(0, a)
R(–a, 0) T(a, 0)O(0, 0)
x
y
F(?, ?)E(?, a)
D(?, ?)x
y
T(a, ?)
R(?, b)
S(?, ?)
© Glencoe/McGraw-Hill 224 Glencoe Geometry
How Many Triangles?Each puzzle below contains many triangles. Count them carefully.Some triangles overlap other triangles.
How many triangles are there in each figure?
1. 2. 3.
4. 5. 6.
How many triangles can you form by joining points on each circle? List the vertices of each triangle.
7. 8.
8. 9. QR
P
U
S
TV
J K
O
L
MN
E F
I
GH
B
C
DA
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Cla
ssify
ing
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill18
3G
lenc
oe G
eom
etry
Lesson 4-1
Cla
ssif
y Tr
ian
gle
s b
y A
ng
les
One
way
to
clas
sify
a t
rian
gle
is b
y th
e m
easu
res
of it
s an
gles
.
•If
one
of th
e an
gles
of a
tria
ngle
is a
n ob
tuse
ang
le, t
hen
the
tria
ngle
is a
n ob
tuse
tri
angl
e.
•If
one
of th
e an
gles
of a
tria
ngle
is a
rig
ht a
ngle
, the
n th
e tr
iang
le is
a r
ight
tri
angl
e.
•If
all t
hree
of th
e an
gles
of a
tria
ngle
are
acu
te a
ngle
s, th
en th
e tr
iang
le is
an
acut
e tr
iang
le.
•If
all t
hree
ang
les
of a
n ac
ute
tria
ngle
are
con
grue
nt, t
hen
the
tria
ngle
is a
n eq
uian
gula
r tr
iang
le.
Cla
ssif
y ea
ch t
rian
gle.
a.
All
thre
e an
gles
are
con
grue
nt,s
o al
l thr
ee a
ngle
s ha
ve m
easu
re 6
0°.
The
tri
angl
e is
an
equi
angu
lar
tria
ngle
.
b.
The
tri
angl
e ha
s on
e an
gle
that
is o
btus
e.It
is a
n ob
tuse
tri
angl
e.
c.
The
tri
angl
e ha
s on
e ri
ght
angl
e.It
is a
rig
ht t
rian
gle.
Cla
ssif
y ea
ch t
rian
gle
as a
cute
,equ
ian
gula
r,ob
tuse
,or
righ
t.
1.2.
3.
righ
tob
tuse
equi
angu
lar
4.5.
6.
acut
eri
ght
obtu
se
60!
28!
92!
FDB
45!
45!
90!
XY
W
65!
65!
50!
UV
T
60!
60!
60!
Q
RS
120!
30!
30!
NO
P
67!
90!
23!
K LM
90!
60!
30!
G
HJ
25!
35!
120!
DF
E
60!A
BC
Exam
ple
Exam
ple
Exercis
esExercis
es
©G
lenc
oe/M
cGra
w-H
ill18
4G
lenc
oe G
eom
etry
Cla
ssif
y Tr
ian
gle
s b
y Si
des
You
can
clas
sify
a t
rian
gle
by t
he m
easu
res
of it
s si
des.
Equ
al n
umbe
rs o
f ha
sh m
arks
indi
cate
con
grue
nt s
ides
.
•If
all t
hree
side
s of
a tr
iang
le a
re c
ongr
uent
, the
n th
e tr
iang
le is
an
equi
late
ral t
rian
gle.
•If
at le
ast t
wo
side
s of
a tr
iang
le a
re c
ongr
uent
, the
n th
e tr
iang
le is
an
isos
cele
s tr
iang
le.
•If
no tw
osi
des
of a
tria
ngle
are
con
grue
nt, t
hen
the
tria
ngle
is a
sca
lene
tri
angl
e.
Cla
ssif
y ea
ch t
rian
gle.
a.b.
c.
Tw
o si
des
are
cong
ruen
t.A
ll th
ree
side
s ar
e T
he t
rian
gle
has
no p
air
The
tri
angl
e is
an
cong
ruen
t.T
he t
rian
gle
of c
ongr
uent
sid
es.I
t is
is
osce
les
tria
ngle
.is
an
equi
late
ral t
rian
gle.
a sc
alen
e tr
iang
le.
Cla
ssif
y ea
ch t
rian
gle
as e
quil
ate
ral,
isos
cele
s,or
sca
len
e.
1.2.
3.
scal
ene
equi
late
ral
scal
ene
4.5.
6.
isos
cele
sis
osce
les
equi
late
ral
7.F
ind
the
mea
sure
of
each
sid
e of
equ
ilate
ral !
RS
Tw
ith
RS
!2x
"2,
ST
!3x
,an
d T
R!
5x#
4.2
8.F
ind
the
mea
sure
of
each
sid
e of
isos
cele
s !
AB
Cw
ith
AB
!B
Cif
AB
!4y
,B
C!
3y"
2,an
d A
C!
3y.
AB
"B
C"
8,A
C"
6
9.F
ind
the
mea
sure
of
each
sid
e of
!A
BC
wit
h ve
rtic
es A
(#1,
5),B
(6,1
),an
d C
(2,#
6).
Cla
ssif
y th
e tr
iang
le.
AB
"B
C"
%65!
,AC
"%
130
!;!
AB
Cis
isos
cele
s.
DE
Fx
xx
8x32
x
32x
B CA
UW
S
1217
19Q
O
MG
KI
18
1818
21
3!
"G
CA
2312
15X
V
TN
RP
LJ
H
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Cla
ssify
ing
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Cla
ssify
ing
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill18
5G
lenc
oe G
eom
etry
Lesson 4-1
Use
a p
rotr
acto
r to
cla
ssif
y ea
ch t
rian
gle
as a
cute
,equ
ian
gula
r,ob
tuse
,or
righ
t.
1.2.
3.
equi
angu
lar
obtu
seri
ght
4.5.
6.
acut
eob
tuse
acut
e
Iden
tify
th
e in
dic
ated
typ
e of
tri
angl
es.
7.ri
ght
8.is
osce
les
!A
BE
,!B
CE
!B
CD
,!B
DE
9.sc
alen
e10
.obt
use
!A
BE
,!B
CE
!B
DE
ALG
EBR
AF
ind
xan
d t
he
mea
sure
of
each
sid
e of
th
e tr
ian
gle.
11.!
AB
Cis
equ
ilate
ral w
ith
AB
!3x
#2,
BC
!2x
"4,
and
CA
!x
"10
.x
"6,
AB
"16
,BC
"16
,CA
"16
12.!
DE
Fis
isos
cele
s,"
Dis
the
ver
tex
angl
e,D
E!
x"
7,D
F!
3x#
1,an
d E
F!
2x"
5.x
"4,
DE
"11
,DF
"11
,EF
"13
Fin
d t
he
mea
sure
s of
th
e si
des
of
!R
ST
and
cla
ssif
y ea
ch t
rian
gle
by i
ts s
ides
.
13.R
(0,2
),S
(2,5
),T
(4,2
)
RS
"%
13!,S
T"
%13!
,RT
"4;
isos
cele
s
14.R
(1,3
),S
(4,7
),T
(5,4
)
RS
"5,
ST
"%
10!,R
T"
%17!
;sca
lene
EC
D
AB
©G
lenc
oe/M
cGra
w-H
ill18
6G
lenc
oe G
eom
etry
Use
a p
rotr
acto
r to
cla
ssif
y ea
ch t
rian
gle
as a
cute
,equ
ian
gula
r,ob
tuse
,or
righ
t.
1.2.
3.
obtu
seac
ute
righ
t
Iden
tify
th
e in
dic
ated
typ
e of
tri
angl
es i
f A !
B!"
A!D!
"B!
D!"
D!C!
,B!E!
"E!
D!,A!
B!⊥
B!C!
,an
d E!
D!⊥
D!C!
.
4.ri
ght
5.ob
tuse
!A
BC
,!C
DE
!B
ED
,!B
DC
6.sc
alen
e7.
isos
cele
s!
AB
C,!
CD
E!
AB
D,!
BE
D,!
BD
C
AL
GE
BR
AF
ind
xan
d t
he
mea
sure
of
each
sid
e of
th
e tr
ian
gle.
8.!
FG
His
equ
ilate
ral w
ith
FG
!x
"5,
GH
!3x
#9,
and
FH
!2x
#2.
x"
7,FG
"12
,GH
"12
,FH
"12
9.!
LM
Nis
isos
cele
s,"
Lis
the
ver
tex
angl
e,L
M!
3x#
2,L
N!
2x"
1,an
d M
N!
5x#
2.x
"3,
LM"
7,LN
"7,
MN
"13
Fin
d t
he
mea
sure
s of
th
e si
des
of
!K
PL
and
cla
ssif
y ea
ch t
rian
gle
by i
ts s
ides
.
10.K
(#3,
2) P
(2,1
),L
(#2,
#3)
KP
"%
26!,P
L"
4%2!,
LK"
%26!
;iso
scel
es
11.K
(5,#
3),P
(3,4
),L
(#1,
1)K
P"
%53!
,PL
"5,
LK"
2%13!
;sca
lene
12.K
(#2,
#6)
,P(#
4,0)
,L(3
,#1)
KP
"2%
10!,P
L"
5%2!,
LK"
5%2!;
isos
cele
s
13.D
ESIG
ND
iana
ent
ered
the
des
ign
at t
he r
ight
in a
logo
con
test
sp
onso
red
by a
wild
life
envi
ronm
enta
l gro
up.U
se a
pro
trac
tor.
How
man
y ri
ght
angl
es a
re t
here
?5
AC
DEB
Pra
ctic
e (A
vera
ge)
Cla
ssify
ing
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csC
lass
ifyin
g Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill18
7G
lenc
oe G
eom
etry
Lesson 4-1
Pre-
Act
ivit
yW
hy
are
tria
ngl
es i
mp
orta
nt
in c
onst
ruct
ion
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
1 at
the
top
of
page
178
in y
our
text
book
.
•W
hy a
re t
rian
gles
use
d fo
r br
aces
in c
onst
ruct
ion
rath
er t
han
othe
r sh
apes
?S
ampl
e an
swer
:Tri
angl
es li
e in
a p
lane
and
are
rig
id s
hape
s.•
Why
do
you
thin
k th
at is
osce
les
tria
ngle
s ar
e us
ed m
ore
ofte
n th
ansc
alen
e tr
iang
les
in c
onst
ruct
ion?
Sam
ple
answ
er:I
sosc
eles
tria
ngle
s ar
e sy
mm
etri
cal.
Rea
din
g t
he
Less
on
1.Su
pply
the
cor
rect
num
bers
to
com
plet
e ea
ch s
ente
nce.
a.In
an
obtu
se t
rian
gle,
ther
e ar
e ac
ute
angl
e(s)
,ri
ght
angl
e(s)
,and
obtu
se a
ngle
(s).
b.In
an
acut
e tr
iang
le,t
here
are
ac
ute
angl
e(s)
,ri
ght
angl
e(s)
,and
obtu
se a
ngle
(s).
c.In
a r
ight
tri
angl
e,th
ere
are
acut
e an
gle(
s),
righ
t an
gle(
s),a
nd
obtu
se a
ngle
(s).
2.D
eter
min
e w
heth
er e
ach
stat
emen
t is
alw
ays,
som
etim
es,o
r ne
ver
true
.a.
A r
ight
tri
angl
e is
sca
lene
.so
met
imes
b.A
n ob
tuse
tri
angl
e is
isos
cele
s.so
met
imes
c.A
n eq
uila
tera
l tri
angl
e is
a r
ight
tri
angl
e.ne
ver
d.A
n eq
uila
tera
l tri
angl
e is
isos
cele
s.al
way
se.
An
acut
e tr
iang
le is
isos
cele
s.so
met
imes
f.A
sca
lene
tri
angl
e is
obt
use.
som
etim
es
3.D
escr
ibe
each
tri
angl
e by
as
man
y of
the
fol
low
ing
wor
ds a
s ap
ply:
acut
e,ob
tuse
,rig
ht,
scal
ene,
isos
cele
s,or
equ
ilat
eral
.a.
b.c.
acut
e,sc
alen
eob
tuse
,iso
scel
esri
ght,
scal
ene
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is t
o re
late
it t
o a
nonm
athe
mat
ical
defi
niti
on o
f th
e sa
me
wor
d.H
ow is
the
use
of
the
wor
d ac
ute,
whe
n us
ed t
o de
scri
beac
ute
pain
,rel
ated
to
the
use
of t
he w
ord
acut
ew
hen
used
to
desc
ribe
an
acut
e an
gle
oran
acu
te t
rian
gle?
Sam
ple
answ
er:B
oth
are
rela
ted
to t
he m
eani
ng o
f ac
ute
as s
harp
.An
acut
e pa
inis
a s
harp
pai
n,an
d an
acu
te a
ngle
can
beth
ough
t of
as
an a
ngle
with
a s
harp
poi
nt.I
n an
acu
te tr
iang
leal
l of
the
angl
es a
re a
cute
.
5
34
135!
80!70!
30!
01
20
03
10
2
©G
lenc
oe/M
cGra
w-H
ill18
8G
lenc
oe G
eom
etry
Rea
ding
Mat
hem
atic
sW
hen
you
read
geo
met
ry,y
ou m
ay n
eed
to d
raw
a d
iagr
am t
o m
ake
the
text
easi
er t
o un
ders
tand
.
Con
sid
er t
hre
e p
oin
ts,A
,B,a
nd
Con
a c
oord
inat
e gr
id.
Th
e y-
coor
din
ates
of
Aan
d B
are
the
sam
e.T
he
x-co
ord
inat
e of
Bis
grea
ter
than
th
e x-
coor
din
ate
of A
.Bot
h c
oord
inat
es o
f C
are
grea
ter
than
th
e co
rres
pon
din
g co
ord
inat
es o
f B
.Is
tria
ngl
e A
BC
acu
te,r
igh
t,or
obt
use
?
To a
nsw
er t
his
ques
tion
,fir
st d
raw
a s
ampl
e tr
iang
le
that
fit
s th
e de
scri
ptio
n.
Side
AB
mus
t be
a h
oriz
onta
l seg
men
t be
caus
e th
e y-
coor
dina
tes
are
the
sam
e.Po
int
Cm
ust
be lo
cate
d to
the
rig
ht a
nd u
p fr
om p
oint
B.
Fro
m t
he d
iagr
am y
ou c
an s
ee t
hat
tria
ngle
AB
Cm
ust
be o
btus
e.
An
swer
eac
h q
ues
tion
.Dra
w a
sim
ple
tri
angl
e on
th
e gr
id a
bove
to
hel
p y
ou.
1.C
onsi
der
thre
e po
ints
,R,S
,and
2.
Con
side
r th
ree
nonc
ollin
ear
poin
ts,
Ton
a c
oord
inat
e gr
id.T
he
J,K
,and
Lon
a c
oord
inat
e gr
id.T
hex-
coor
dina
tes
of R
and
Sar
e th
ey-
coor
dina
tes
of J
and
Kar
e th
esa
me.
The
y-c
oord
inat
e of
Tis
sam
e.T
he x
-coo
rdin
ates
of K
and
Lbe
twee
n th
e y-
coor
dina
tes
of R
are
the
sam
e.Is
tri
angl
e JK
L a
cute
,an
d S
.The
x-c
oord
inat
e of
Tis
less
righ
t,or
obt
use?
righ
tth
an t
he x
-coo
rdin
ate
of R
.Is
angl
eR
of t
rian
gle
RS
T a
cute
,rig
ht,o
r ob
tuse
?ac
ute
3.C
onsi
der
thre
e no
ncol
linea
r po
ints
,4.
Con
side
r th
ree
poin
ts,G
,H,a
nd I
D,E
,and
Fon
a c
oord
inat
e gr
id.
on a
coo
rdin
ate
grid
.Poi
nts
Gan
d T
he x
-coo
rdin
ates
of D
and
Ear
eH
are
on t
he p
osit
ive
y-ax
is,a
ndop
posi
tes.
The
y-c
oord
inat
es o
f Dan
dth
e y-
coor
dina
te o
f Gis
tw
ice
the
Ear
e th
e sa
me.
The
x-c
oord
inat
e of
y-co
ordi
nate
of H
.Poi
nt I
is o
n th
e F
is 0
.Wha
t ki
nd o
f tr
iang
le m
ust
posi
tive
x-a
xis,
and
the
x-co
ordi
nate
!D
EF
be:
scal
ene,
isos
cele
s,or
of I
is g
reat
er t
han
the
y-co
ordi
nate
equi
late
ral?
isos
cele
sof
G.I
s tr
iang
le G
HI
scal
ene,
isos
cele
s,or
equ
ilate
ral?
scal
eneB
A
Q x
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-1
4-1
Exam
ple
Exam
ple
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Ang
les
of T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill18
9G
lenc
oe G
eom
etry
Lesson 4-2
An
gle
Su
m T
heo
rem
If t
he m
easu
res
of t
wo
angl
es o
f a
tria
ngle
are
kno
wn,
the
mea
sure
of
the
thir
d an
gle
can
alw
ays
be f
ound
.
Ang
le S
umT
he s
um o
f the
mea
sure
s of
the
angl
es o
f a tr
iang
le is
180
.Th
eore
mIn
the
figur
e at
the
right
, m"
A"
m"
B"
m"
C!
180.
CA
B
Fin
d m
"T
.
m"
R"
m"
S"
m"
T!
180
Ang
le S
umT
heor
em
25 "
35 "
m"
T!
180
Sub
stitu
tion
60 "
m"
T!
180
Add
.
m"
T!
120
Sub
trac
t 60
from
eac
h si
de.
35!
25!
RT
S
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
m"
1 "
m"
A"
m"
B!
180
Ang
le S
um T
heor
em
m"
1 "
58 "
90!
180
Sub
stitu
tion
m"
1 "
148
!18
0A
dd.
m"
1!
32S
ubtr
act 1
48 fr
omea
ch s
ide.
m"
2!
32V
ertic
al a
ngle
s ar
eco
ngru
ent.
m"
3 "
m"
2 "
m"
E!
180
Ang
le S
um T
heor
em
m"
3 "
32 "
108
!18
0S
ubst
itutio
n
m"
3 "
140
!18
0A
dd.
m"
3!
40S
ubtr
act 1
40 fr
omea
ch s
ide.
58!90
!
108!
12
3
E
DA
C
B
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exercis
esExercis
es
Fin
d t
he
mea
sure
of
each
nu
mbe
red
an
gle.
1.m
"1
"28
2.m
"1
"12
0
3.4.
5.6.
m"
1 "
820
! 152!
DG
A
1
m"
1 "
30,
m"
2 "
6030
!60
!12
S
R
TW
m"
1 "
56,
m"
2 "
56,
m"
3 "
74Q
O
NM P
58!
66!
50!
32
1
m"
1 "
30,
m"
2 "
60V W
T
U
30!
60!
2
1
S
QR
30!
1
90!
62!
1NM
P
©G
lenc
oe/M
cGra
w-H
ill19
0G
lenc
oe G
eom
etry
Exte
rio
r A
ng
le T
heo
rem
At
each
ver
tex
of a
tri
angl
e,th
e an
gle
form
ed b
y on
e si
dean
d an
ext
ensi
on o
f th
e ot
her
side
is c
alle
d an
ext
erio
r an
gle
of t
he t
rian
gle.
For
each
exte
rior
ang
le o
f a
tria
ngle
,the
rem
ote
inte
rior
an
gles
are
the
inte
rior
ang
les
that
are
not
adja
cent
to
that
ext
erio
r an
gle.
In t
he d
iagr
am b
elow
,"B
and
"A
are
the
rem
ote
inte
rior
angl
es f
or e
xter
ior
"D
CB
.
Ext
erio
r A
ngle
The
mea
sure
of a
n ex
terio
r an
gle
of a
tria
ngle
is e
qual
to
Theo
rem
the
sum
of t
he m
easu
res
of th
e tw
o re
mot
e in
terio
r an
gles
.m
"1
!m
"A
"m
"B
AC
B
D1
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Ang
les
of T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
Fin
d m
"1.
m"
1!
m"
R"
m"
SE
xter
ior A
ngle
The
orem
!60
"80
Sub
stitu
tion
!14
0A
dd.
RT
S
60!80
!
1
Fin
d x
.
m"
PQ
S!
m"
R"
m"
SE
xter
ior A
ngle
The
orem
78 !
55 "
xS
ubst
itutio
n
23 !
xS
ubtr
act 5
5 fr
om e
ach
side
.
SR
Q
P
55!
78!
x!
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exercis
esExercis
es
Fin
d t
he
mea
sure
of
each
nu
mbe
red
an
gle.
1.2.
m"
1 "
115
m"
1 "
60,m
"2
"12
0
3.4.
m"
1 "
60,m
"2
"60
,m"
3 "
120
m"
1 "
109,
m"
2 "
29,m
"3
"71
Fin
d x
.
5.25
6.29
E
FG
H58
!x!
x!B
A
DC
95!
2 x!
145!
UT
SR
V
35!
36!
80!
13
2
PO
Q
NM
60! 60
!3
2
1
BC
D
A
25!
35! 1
2Y
ZW
X 65!50
!
1
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Ang
les
of T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill19
1G
lenc
oe G
eom
etry
Lesson 4-2
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
1.27
2.17
,17
Fin
d t
he
mea
sure
of
each
an
gle.
3.m
"1
55
4.m
"2
55
5.m
"3
70
Fin
d t
he
mea
sure
of
each
an
gle.
6.m
"1
125
7.m
"2
55
8.m
"3
95
Fin
d t
he
mea
sure
of
each
an
gle.
9.m
"1
140
10.m
"2
40
11.m
"3
65
12.m
"4
75
13.m
"5
115
Fin
d t
he
mea
sure
of
each
an
gle.
14.m
"1
27
15.m
"2
2763
!
1
2D
AC
B
80!
60!
40!
105!
14
52
3
150!
55!
70!
12
3
85!
55!
40!
12
3
146!
TIG
ERS
80! 73!
©G
lenc
oe/M
cGra
w-H
ill19
2G
lenc
oe G
eom
etry
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
1.18
2.85
Fin
d t
he
mea
sure
of
each
an
gle.
3.m
"1
97
4.m
"2
83
5.m
"3
62
Fin
d t
he
mea
sure
of
each
an
gle.
6.m
"1
104
7.m
"4
45
8.m
"3
65
9.m
"2
79
10.m
"5
73
11.m
"6
147
Fin
d t
he
mea
sure
of
each
an
gle
if "
BA
Dan
d
"B
DC
are
righ
t an
gles
an
d m
"A
BC
"84
.
12.m
"1
26
13.m
"2
32
14.C
ON
STR
UC
TIO
NT
he d
iagr
am s
how
s an
ex
ampl
e of
the
Pra
tt T
russ
use
d in
bri
dge
cons
truc
tion
.Use
the
dia
gram
to
find
m"
1.55
145!
1
64!
1
2AB
C
D
118!
36!
68!
70! 65
! 82!
1
2
34
5
6
58!
39!
35!
12
3
40!
55!
72!
?
Pra
ctic
e (A
vera
ge)
Ang
les
of T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csA
ngle
s of
Tri
angl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill19
3G
lenc
oe G
eom
etry
Lesson 4-2
Pre-
Act
ivit
yH
ow a
re t
he
angl
es o
f tr
ian
gles
use
d t
o m
ake
kit
es?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
2 at
the
top
of
page
185
in y
our
text
book
.
The
fra
me
of t
he s
impl
est
kind
of
kite
div
ides
the
kit
e in
to f
our
tria
ngle
s.D
escr
ibe
thes
e fo
ur t
rian
gles
and
how
the
y ar
e re
late
d to
eac
h ot
her.
Sam
ple
answ
er:T
here
are
tw
o pa
irs
of r
ight
tri
angl
es t
hat
have
the
sam
e si
ze a
nd s
hape
.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
thre
e in
teri
or a
ngle
s of
the
tri
angl
e.(U
se t
hree
lett
ers
to n
ame
each
ang
le.)
"B
AC
,"A
BC
,"B
CA
b.N
ame
thre
e ex
teri
or a
ngle
s of
the
tri
angl
e.(U
se t
hree
lett
ers
to n
ame
each
ang
le.)
"E
AB
,"D
BC
,"FC
Ac.
Nam
e th
e re
mot
e in
teri
or a
ngle
s of
"E
AB
."
AB
C,"
BC
Ad.
Fin
d th
e m
easu
re o
f ea
ch a
ngle
wit
hout
usi
ng a
pro
trac
tor.
i."
DB
C62
ii.
"A
BC
118
iii.
"A
CF
157
iv.
"E
AB
141
2.In
dica
te w
heth
er e
ach
stat
emen
t is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,rep
lace
the
unde
rlin
ed w
ord
or n
umbe
r w
ith
a w
ord
or n
umbe
r th
at w
ill m
ake
the
stat
emen
t tr
ue.
a.T
he a
cute
ang
les
of a
rig
ht t
rian
gle
are
.fa
lse;
com
plem
enta
ryb.
The
sum
of
the
mea
sure
s of
the
ang
les
of a
ny t
rian
gle
is
.fa
lse;
180
c.A
tri
angl
e ca
n ha
ve a
t m
ost
one
righ
t an
gle
or
angl
e.fa
lse;
obtu
sed.
If t
wo
angl
es o
f on
e tr
iang
le a
re c
ongr
uent
to
two
angl
es o
f an
othe
r tr
iang
le,t
hen
the
thir
d an
gles
of
the
tria
ngle
s ar
e .
true
e.T
he m
easu
re o
f an
ext
erio
r an
gle
of a
tri
angl
e is
equ
al t
o th
e of
the
mea
sure
s of
the
tw
o re
mot
e in
teri
or a
ngle
s.fa
lse;
sum
f.If
the
mea
sure
s of
tw
o an
gles
of
a tr
iang
le a
re 6
2 an
d 93
,the
n th
e m
easu
re o
f th
eth
ird
angl
e is
.
fals
e;25
g.A
n an
gle
of a
tri
angl
e fo
rms
a lin
ear
pair
wit
h an
inte
rior
ang
le o
f th
etr
iang
le.
true
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stud
ents
rem
embe
r m
athe
mat
ical
idea
s an
d fa
cts
mor
e ea
sily
if t
hey
see
them
dem
onst
rate
d vi
sual
ly r
athe
r th
an h
avin
g th
em s
tate
d in
wor
ds.D
escr
ibe
a vi
sual
way
to d
emon
stra
te t
he A
ngle
Sum
The
orem
.S
ampl
e an
swer
:Cut
off
the
ang
les
of a
tri
angl
e an
d pl
ace
them
si
de-b
y-si
de o
n on
e si
de o
f a
line
so t
hat
thei
r ve
rtic
es m
eet
at a
com
mon
poin
t.Th
e re
sult
will
sho
w t
hree
ang
les
who
se m
easu
res
add
up t
o 18
0.
exte
rior
35
diff
eren
ce
cong
ruen
t
acut
e
100
supp
lem
enta
ry
39!
23!
EA
BD
CF
©G
lenc
oe/M
cGra
w-H
ill19
4G
lenc
oe G
eom
etry
Find
ing
Ang
le M
easu
res
in T
rian
gles
You
can
use
alge
bra
to s
olve
pro
blem
s in
volv
ing
tria
ngle
s.
In t
rian
gle
AB
C,m
"A
,is
twic
e m
"B
,an
d m
"C
is 8
mor
e th
an m
"B
.Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
Wri
te a
nd s
olve
an
equa
tion
.Let
x!
m"
B.
m"
A"
m"
B"
m"
C!
180
2x"
x"
(x"
8)!
180
4x"
8!
180
4x!
172
x!
43So
,m"
A!
2(43
)or
86,m
"B
!43
,and
m"
C!
43"
8or
51.
Sol
ve e
ach
pro
blem
.
1.In
tri
angl
e D
EF
,m"
E is
thr
ee t
imes
2.In
tri
angl
e R
ST
,m"
Tis
5 m
ore
than
m
"D
,and
m"
Fis
9 le
ss t
han
m"
E.
m"
R,a
nd m
"S
is 1
0 le
ss t
han
m"
T.
Wha
t is
the
mea
sure
of
each
ang
le?
Wha
t is
the
mea
sure
of
each
ang
le?
m"
D"
27,m
"E
"81
,m"
F"
72m
"R
"60
,m"
S"
55,m
"T
"65
3.In
tri
angl
e JK
L,m
"K
is f
our
tim
es4.
In t
rian
gle
XY
Z,m
"Z
is 2
mor
e th
an t
wic
em
"J,
and
m"
Lis
fiv
e ti
mes
m"
J.m
"X
,and
m"
Yis
7 le
ss t
han
twic
e m
"X
.W
hat
is t
he m
easu
re o
f ea
ch a
ngle
?W
hat
is t
he m
easu
re o
f ea
ch a
ngle
?
m"
J"
18,m
"K
"72
,m"
L"
90m
"X
"37
,m"
Y"
67,m
"Z
"76
5.In
tri
angl
e G
HI,
m"
H is
20
mor
e th
an6.
In t
rian
gle
MN
O,m
"M
is e
qual
to
m"
N,
m"
G,a
nd m
"G
is 8
mor
e th
an m
"I.
and
m"
Ois
5 m
ore
than
thr
ee t
imes
W
hat
is t
he m
easu
re o
f ea
ch a
ngle
?m
"N
.Wha
t is
the
mea
sure
of
each
ang
le?
m"
G"
56,m
"H
"76
,m"
I"48
m"
M"
m"
N"
35,m
"O
"11
0
7.In
tri
angl
e S
TU
,m"
U is
hal
f m
"T
,8.
In t
rian
gle
PQ
R,m
"P
is e
qual
to
and
m"
Sis
30
mor
e th
an m
"T
.Wha
tm
"Q
,and
m"
Ris
24
less
tha
n m
"P.
is t
he m
easu
re o
f ea
ch a
ngle
?W
hat
is t
he m
easu
re o
f ea
ch a
ngle
?
m"
S"
90,m
"T
"60
,m"
U"
30m
"P
"m
"Q
"68
,m"
R"
44
9.W
rite
you
r ow
n pr
oble
ms
abou
t m
easu
res
of t
rian
gles
.S
ee s
tude
nts’
wor
k.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-2
4-2
Exam
ple
Exam
ple
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Con
grue
nt T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
5G
lenc
oe G
eom
etry
Lesson 4-3
Co
rres
po
nd
ing
Par
ts o
f C
on
gru
ent
Tria
ng
les
Tri
angl
es t
hat
have
the
sam
e si
ze a
nd s
ame
shap
e ar
e co
ngr
uen
t tr
ian
gles
.Tw
o tr
iang
les
are
cong
ruen
t if
and
on
ly if
all
thre
e pa
irs
of c
orre
spon
ding
ang
les
are
cong
ruen
t an
d al
l thr
ee p
airs
of
corr
espo
ndin
g si
des
are
cong
ruen
t.In
th
e fi
gure
,!A
BC
#!
RS
T.
If !
XY
Z"
!R
ST
,nam
e th
e p
airs
of
con
gru
ent
angl
es a
nd
con
gru
ent
sid
es.
"X
#"
R,"
Y#
"S
,"Z
#"
TX $
Y$#
R$S$,
X$Z$
#R$
T$,Y$
Z$#
S$T$
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
3.
!A
BC
"!
JKL
!A
BC
"!
DC
B!
JKM
"!
LMK
Nam
e th
e co
rres
pon
din
g co
ngr
uen
t an
gles
an
d s
ides
for
th
e co
ngr
uen
t tr
ian
gles
.
4.5.
6.
"E
""
J;"
F"
"K
;"
A"
"D
;"
R"
"T;
"G
""
L;E!F!
"J!K!
;"
AB
C"
"D
CB
;"
RS
U"
"TS
U;
E!G!"
J!L!;F!
G!"
K!L!
"A
CB
""
DB
C;
"R
US
""
TUS
;A!
B!"
D!C!
;A!C!
"D!
B!;
R!U!
"T!U!
;R!S!
"T!S!
;B!
C!"
C!B!
S!U!"
S!U!
R TUS
BD
CA
FG
LK J
E
K J
L MC
D
A
B
CA
B
LJK
Y
XZ
T
SR
AC
BR
TS
Exam
ple
Exam
ple
Exercis
esExercis
es
©G
lenc
oe/M
cGra
w-H
ill19
6G
lenc
oe G
eom
etry
Iden
tify
Co
ng
ruen
ce T
ran
sfo
rmat
ion
sIf
tw
o tr
iang
les
are
cong
ruen
t,yo
u ca
nsl
ide,
flip
,or
turn
one
of
the
tria
ngle
s an
d th
ey w
ill s
till
be c
ongr
uent
.The
se a
re c
alle
dco
ngr
uen
ce t
ran
sfor
mat
ion
sbe
caus
e th
ey d
o no
t ch
ange
the
siz
e or
sha
pe o
f th
e fi
gure
.It
is c
omm
on t
o us
e pr
ime
sym
bols
to
dist
ingu
ish
betw
een
an o
rigi
nal !
AB
Can
d a
tran
sfor
med
!A
$B$C
$.
Nam
e th
e co
ngr
uen
ce t
ran
sfor
mat
ion
th
at p
rod
uce
s !
A#B
#C#
from
!A
BC
.T
he c
ongr
uenc
e tr
ansf
orm
atio
n is
a s
lide.
"A
#"
A$;
"B
#"
B$;
"C
#"
C$;
A $B$
#A$
$$B$$$;
A$C$
#A$
$$C$$$;
B$C$
#B$
$$C$$$
Des
crib
e th
e co
ngr
uen
ce t
ran
sfor
mat
ion
bet
wee
n t
he
two
tria
ngl
es a
s a
slid
e,a
flip
,or
a tu
rn.T
hen
nam
e th
e co
ngr
uen
t tr
ian
gles
.
1.2.
flip;
!R
ST
"!
RS
#T#
slid
e;!
MN
P"
!M
#N#P
#
3.4.
turn
;!O
PQ
"!
OP
#Q#
flip;
!A
BC
"!
AB
#C
5.6.
slid
e;!
AB
C"
!A
#B#C
#tu
rn;!
MN
P"
!M
N#P
#
x
y
OM
N
P
N$
P$
x
y
OA
$B
$
C$
ABC
x
y
OB
$B
A Cx
y
O
Q$
P$
P Q
x
y
ON
$
M$
P$
N MP
x
y
OR
S$
T$
S
T
x
y
O
A$
B$
B
C$
AC
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Con
grue
nt T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Con
grue
nt T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
7G
lenc
oe G
eom
etry
Lesson 4-3
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
!JP
L"
!TV
S!
AB
C"
!W
XY
3.4.
!P
QR
"!
PS
R!
DE
F"
!D
GF
Nam
e th
e co
ngr
uen
t an
gles
an
d s
ides
for
eac
h p
air
of c
ongr
uen
t tr
ian
gles
.
5.!
AB
C#
!F
GH
"A
""
F,"
B"
"G
,"C
""
H;A!
B!"
F!G!,B!
C!"
G!H!
,A!C!
"F!H!
6.!
PQ
R#
!S
TU
"P
""
S,"
Q"
"T,
"R
""
U;P!
Q!"
S!T!,Q!
R!"
T!U!,P!
R!"
S!U!
Ver
ify
that
eac
h o
f th
e fo
llow
ing
tran
sfor
mat
ion
s p
rese
rves
con
gru
ence
,an
d n
ame
the
con
gru
ence
tra
nsf
orm
atio
n.
7.!
AB
C#
!A
$B$C
$8.
!D
EF
#!
D$E
$F$
AB
"2%
2!,A
#B#
"2%
2!,D
E"
4,D
#E#
"4,
EF
"5,
BC
"2%
2!,B
#C#
"2%
2!,E
#F#
"5,
DF
" 3
,D#F
#"
3,
AC
"4,
A#C
#"
4,"
A"
"A
#,"
D"
"D
#,"
E"
"E
#,
"B
""
B#,
"C
""
C#;
slid
e"
F"
"F
#;fli
p
x
y
OD
$
E$
F$
DE
F
x
y
OA$
B$
C$
A
B
C
D
E G
FR
P
Q S
WY
XC
AB
L
P
J
S
V
T
©G
lenc
oe/M
cGra
w-H
ill19
8G
lenc
oe G
eom
etry
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
!A
BC
"!
DR
S!
LMN
"!
QP
N
Nam
e th
e co
ngr
uen
t an
gles
an
d s
ides
for
eac
h p
air
of c
ongr
uen
t tr
ian
gles
.
3.!
GK
P#
!L
MN
"G
""
L,"
K"
"M
,"P
""
N;G!
K!"
L!M!,K!
P!"
M!N!
,G!P!
"L!N!
4.!
AN
C#
!R
BV
"A
""
R,"
N"
"B
,"C
""
V;A!
N!"
R!B!
,N!C!
"B!
V!,A!
C!"
R!V!
Ver
ify
that
eac
h o
f th
e fo
llow
ing
tran
sfor
mat
ion
s p
rese
rves
con
gru
ence
,an
d n
ame
the
con
gru
ence
tra
nsf
orm
atio
n.
5.!
PS
T#
!P
$S$T
$6.
!L
MN
#!
L$M
$N$
PS
"%
13!,P
#S#
"%
13!,
LM"
2%2!,
L#M
#"
2%2!,
ST
"%
5!,S
#T#
"%
5!,P
T"
%10!
,M
N"
%29!
,M#N
#"
%29!
,P
#T#
"%
10!,"
P"
"P
#,LN
"7,
L#N
#"
7,"
L"
"L#
,"
S"
"S
#,"
T"
!T
#;fli
p"
M"
"M
#,"
N"
"N
#;fli
p
QU
ILTI
NG
For
Exe
rcis
es 7
an
d 8
,ref
er t
o th
e qu
ilt
des
ign
.
7.In
dica
te t
he t
rian
gles
tha
t ap
pear
to
be c
ongr
uent
.!
AB
I"!
EB
F,!
CB
D"
!H
BG
8.N
ame
the
cong
ruen
t an
gles
and
con
grue
nt s
ides
of
a pa
ir o
f co
ngru
ent
tria
ngle
s.S
ampl
e an
swer
:"A
""
E,"
AB
I""
EB
F,"
I""
F;A!
B!"
E!B!,B!
I!"B!
F!,A!
I!"E!F!
B
A I
E FH
G
CD
x
y
O
M$
N$
L$
M
NL
x
y
O
S$
T$
P$
S
TP
MN
L
P
QD
R
SC
A
BPra
ctic
e (A
vera
ge)
Con
grue
nt T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
Answers (Lesson 4-3)
©G
lenc
oe/M
cGra
w-H
ill20
0G
lenc
oe G
eom
etry
Tran
sfor
mat
ions
in T
he C
oord
inat
e P
lane
The
fol
low
ing
stat
emen
t te
lls o
ne w
ay t
o m
ap p
reim
age
poin
ts t
o im
age
poin
ts in
the
coo
rdin
ate
plan
e.
(x,y
) →(x
"6,
y#
3)
Thi
s ca
n be
rea
d,“T
he p
oint
wit
h co
ordi
nate
s (x
,y)
is
map
ped
to t
he p
oint
wit
h co
ordi
nate
s (x
"6,
y#
3).”
Wit
h th
is t
rans
form
atio
n,fo
r ex
ampl
e,(3
,5)
is m
appe
d to
(3
"6,
5 #
3) o
r (9
,2).
The
fig
ure
show
s ho
w t
he t
rian
gle
AB
Cis
map
ped
to t
rian
gle
XY
Z.
1.D
oes
the
tran
sfor
mat
ion
abov
e ap
pear
to
be a
con
grue
nce
tran
sfor
mat
ion?
Exp
lain
you
ran
swer
.Ye
s;th
e tr
ansf
orm
atio
n sl
ides
the
fig
ure
to t
he lo
wer
rig
ht w
ithou
tch
angi
ng it
s si
ze o
r sh
ape.
Dra
w t
he
tran
sfor
mat
ion
im
age
for
each
fig
ure
.Th
en t
ell
wh
eth
er t
he
tran
sfor
mat
ion
is
or i
s n
ot a
con
gru
ence
tra
nsf
orm
atio
n.
2.(x
,y) →
(x#
4,y)
yes
3.(x
,y) →
(x"
8,y
"7)
yes
4.(x
,y) →
(#x
,#y)
yes
5.(x
,y) →
%#%1 2% x
,y&
no
x
y
Ox
y
O
x
y
Ox
y
O
x
y
B
A
CX
ZY
O
(x, y
) → (x
$ 6
, y %
3)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csC
ongr
uent
Tri
angl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
9G
lenc
oe G
eom
etry
Lesson 4-3
Pre-
Act
ivit
yW
hy
are
tria
ngl
es u
sed
in
bri
dge
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
3 at
the
top
of
page
192
in y
our
text
book
.
In t
he b
ridg
e sh
own
in t
he p
hoto
grap
h in
you
r te
xtbo
ok,d
iago
nal b
race
sw
ere
used
to
divi
de s
quar
es in
to t
wo
isos
cele
s ri
ght
tria
ngle
s.W
hy d
o yo
uth
ink
thes
e br
aces
are
use
d on
the
bri
dge?
Sam
ple
answ
er:T
hedi
agon
al b
race
s m
ake
the
stru
ctur
e st
rong
er a
nd p
reve
nt it
from
bei
ng d
efor
med
whe
n it
has
to w
ithst
and
a he
avy
load
.
Rea
din
g t
he
Less
on
1.If
!R
ST
#!
UW
V,c
ompl
ete
each
pai
r of
con
grue
nt p
arts
.
"R
##
"W
"T
#
R$T$
##
U$W$
#W$
V$
2.Id
enti
fy t
he c
ongr
uent
tri
angl
es in
eac
h di
agra
m.
a.!
AB
C"
!A
DC
b.!
PQ
S"
!R
QS
c.d.
!M
NO
"!
QP
O!
RTV
"!
US
V3.
Det
erm
ine
whe
ther
eac
h st
atem
ent
says
tha
t co
ngru
ence
of
tria
ngle
s is
ref
lexi
ve,
sym
met
ric,
or t
rans
itiv
e.a.
If t
he f
irst
of
two
tria
ngle
s is
con
grue
nt t
o th
e se
cond
tri
angl
e,th
en t
he s
econ
dtr
iang
le is
con
grue
nt t
o th
e fi
rst.
sym
met
ric
b.If
the
re a
re t
hree
tri
angl
es fo
r w
hich
the
firs
t is
con
grue
nt t
o th
e se
cond
and
the
sec
ond
is c
ongr
uent
to
the
thir
d,th
en t
he f
irst
tri
angl
e is
con
grue
nt t
o th
e th
ird.
tran
sitiv
ec.
Eve
ry t
rian
gle
is c
ongr
uent
to
itse
lf.re
flexi
ve
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hing
is t
o ex
plai
n it
to
som
eone
els
e.Yo
ur c
lass
mat
e B
en is
havi
ng t
roub
le w
riti
ng c
ongr
uenc
e st
atem
ents
for
tri
angl
es b
ecau
se h
e th
inks
he
has
tom
atch
up
thre
e pa
irs
of s
ides
and
thr
ee p
airs
of a
ngle
s.H
ow c
an y
ou h
elp
him
und
erst
and
how
to
wri
te c
orre
ct c
ongr
uenc
e st
atem
ents
mor
e ea
sily
?S
ampl
e an
swer
:Wri
te t
heth
ree
vert
ices
of
one
tria
ngle
in a
ny o
rder
.The
n w
rite
the
cor
resp
ondi
ngve
rtic
es o
f th
e se
cond
tri
angl
e in
the
sam
e or
der.
If th
e an
gles
are
wri
tten
in t
he c
orre
ct c
orre
spon
denc
e,th
e si
des
will
aut
omat
ical
ly b
e in
the
corr
ect
corr
espo
nden
ce a
lso.
RT
US
V
NO
P
QM
S
PR
Q
CA
B D
S!T!R!
S!U!
V!
"V
"S
"U
Answers (Lesson 4-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Pro
ving
Con
grue
nce—
SS
S,S
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
1G
lenc
oe G
eom
etry
Lesson 4-4
SSS
Post
ula
teYo
u kn
ow t
hat
two
tria
ngle
s ar
e co
ngru
ent
if c
orre
spon
ding
sid
es a
reco
ngru
ent
and
corr
espo
ndin
g an
gles
are
con
grue
nt.T
he S
ide-
Side
-Sid
e (S
SS)
Post
ulat
e le
tsyo
u sh
ow t
hat
two
tria
ngle
s ar
e co
ngru
ent
if y
ou k
now
onl
y th
at t
he s
ides
of
one
tria
ngle
are
cong
ruen
t to
the
sid
es o
f th
e se
cond
tri
angl
e.
SS
S P
ostu
late
If th
e si
des
of o
ne tr
iang
le a
re c
ongr
uent
to th
e si
des
of a
sec
ond
tria
ngle
, th
en th
e tr
iang
les
are
cong
ruen
t.
Wri
te a
tw
o-co
lum
n p
roof
.G
iven
:A $
B$#
D$B$
and
Cis
the
mid
poin
t of
A$D$
.P
rove
:!A
BC
#!
DB
C
Stat
emen
tsR
easo
ns
1.A$
B$#
D$B$
1.G
iven
2.C
is t
he m
idpo
int
of A$
D$.
2.G
iven
3.A$
C$#
D$C$
3.D
efin
itio
n of
mid
poin
t4.
B$C$
#B$
C$4.
Ref
lexi
ve P
rope
rty
of #
5.!
AB
C#
!D
BC
5.SS
S Po
stul
ate
Wri
te a
tw
o-co
lum
n p
roof
.
B CD
A
Exam
ple
Exam
ple
Exercis
esExercis
es
1.
Giv
en:A$
B$#
X$Y$
,A$C$
#X$
Z$,B$
C$#
Y$Z$
Pro
ve:!
AB
C#
!X
YZ
Stat
emen
tsR
easo
ns
1.A!
B!"
X!Y!1.
Giv
enA!
C!"
X!Z!B!
C!"
Y!Z!2.
!A
BC
"!
XY
Z2.
SS
S P
ost.
2.
Giv
en:R$
S$#
U$T$
,R$T$
#U$
S$P
rove
:!R
ST
#!
UT
S
Stat
emen
tsR
easo
ns
1.R!
S!"
U!T!
1.G
iven
R!T!
"U!
S!2.
S!T!"
T!S!2.
Ref
l.P
rop.
3.!
RS
T"
!U
TS3.
SS
S P
ost.
TU
RS
BY
CA
XZ
©G
lenc
oe/M
cGra
w-H
ill20
2G
lenc
oe G
eom
etry
SAS
Post
ula
teA
noth
er w
ay t
o sh
ow t
hat
two
tria
ngle
s ar
e co
ngru
ent
is t
o us
e th
e Si
de-A
ngle
-Sid
e (S
AS)
Pos
tula
te.
SA
S P
ostu
late
If tw
o si
des
and
the
incl
uded
ang
le o
f one
tria
ngle
are
con
grue
nt to
two
side
s an
d th
e in
clud
ed a
ngle
of a
noth
er tr
iang
le, t
hen
the
tria
ngle
s ar
e co
ngru
ent.
For
eac
h d
iagr
am,d
eter
min
e w
hic
h p
airs
of
tria
ngl
es c
an b
ep
rove
d c
ongr
uen
t by
th
e S
AS
Pos
tula
te.
a.b.
c.
In !
AB
C,t
he a
ngle
is n
ot
The
rig
ht a
ngle
s ar
e T
he in
clud
ed a
ngle
s,"
1 “i
nclu
ded”
by t
he s
ides
A$B$
cong
ruen
t an
d th
ey a
re t
hean
d "
2,ar
e co
ngru
ent
an
d A $
C$.S
o th
e tr
iang
les
incl
uded
ang
les
for
the
beca
use
they
are
ca
nnot
be
prov
ed c
ongr
uent
co
ngru
ent
side
s.al
tern
ate
inte
rior
ang
les
by t
he S
AS
Post
ulat
e.!
DE
F#
!JG
Hby
the
fo
r tw
o pa
ralle
l lin
es.
SAS
Post
ulat
e.!
PS
R#
!R
QP
by t
he
SAS
Post
ulat
e.
For
eac
h f
igu
re,d
eter
min
e w
hic
h p
airs
of
tria
ngl
es c
an b
e p
rove
d c
ongr
uen
t by
the
SA
S P
ostu
late
.
1.2.
3.
!TR
U"
!P
MN
by t
he
"X
QY
and
"W
QZ
are
"M
PL
""
NP
LS
AS
Pos
tula
te.
not
the
incl
uded
ang
les
beca
use
both
are
fo
r th
e co
ngru
ent
righ
t an
gles
.se
gmen
ts.T
he t
rian
gles
!
MP
L"
!N
PL
by
are
not
cong
ruen
t by
th
e S
AS
Pos
tula
te.
the
SA
S P
ostu
late
.
4.5.
6.
The
tria
ngle
s ca
nnot
"
D"
"B
beca
use
The
cong
ruen
t be
pro
ved
cong
ruen
t bo
th a
re r
ight
ang
les.
angl
es a
re t
he
by t
he S
AS
Pos
tula
te.
The
two
tria
ngle
s ar
e in
clud
ed a
ngle
s fo
r co
ngru
ent
by t
he S
AS
th
e co
ngru
ent
side
s.P
ostu
late
.!
FJH
"!
GH
Jby
the
SA
S P
ostu
late
.
JH
GF
K
CBA D
V
T
W
M
PL
N M
X
WZY
QT
P
UN
MR
PQ
S
1
2R
DH
FE
G
J
A BC
X YZ
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Pro
ving
Con
grue
nce—
SS
S,S
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Pro
ving
Con
grue
nce—
SS
S,S
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
3G
lenc
oe G
eom
etry
Lesson 4-4
Det
erm
ine
wh
eth
er !
AB
C"
!K
LM
give
n t
he
coor
din
ates
of
the
vert
ices
.Exp
lain
.
1.A
(#3,
3),B
(#1,
3),C
(#3,
1),K
(1,4
),L
(3,4
),M
(1,6
)
AB
"2,
KL
"2,
BC
"2%
2!,LM
"2%
2!,A
C"
2,K
M"
2.Th
e co
rres
pond
ing
side
s ha
ve t
he s
ame
mea
sure
and
are
con
grue
nt,
so !
AB
C"
!K
LMby
SS
S.
2.A
(#4,
#2)
,B(#
4,1)
,C(#
1,#
1),K
(0,#
2),L
(0,1
),M
(4,1
)
AB
"3,
KL
"3,
BC
"%
13!,L
M"
4,A
C"
%10!
,KM
"5.
The
corr
espo
ndin
g si
des
are
not
cong
ruen
t,so
!A
BC
is n
ot
cong
ruen
t to
!K
LM.
3.W
rite
a f
low
pro
of.
Giv
en:
P $R$#
D$E$
,P$T$
#D$
F$"
R#
"E
,"T
#"
FP
rove
:!
PR
T#
!D
EF
Pro
of:
Det
erm
ine
wh
ich
pos
tula
te c
an b
e u
sed
to
pro
ve t
hat
th
e tr
ian
gles
are
con
gru
ent.
If i
t is
not
pos
sibl
e to
pro
ve t
hat
th
ey a
re c
ongr
uen
t,w
rite
not
pos
sibl
e.
4.5.
6.
SS
SS
AS
not
poss
ible
PR "
DE
Give
n
PT "
DF
Give
n
"R
" "
E Gi
ven
"P
" "
D
Third
Ang
leTh
eore
m
!PR
T "
!D
EF
SAS
"T
" "
F Gi
ven
T
R
P
F
E
D
©G
lenc
oe/M
cGra
w-H
ill20
4G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er !
DE
F"
!P
QR
give
n t
he
coor
din
ates
of
the
vert
ices
.Exp
lain
.
1.D
(#6,
1),E
(1,2
),F
(#1,
#4)
,P(0
,5),
Q(7
,6),
R(5
,0)
DE
"5%
2!,P
Q"
5%2!,
EF
"2%
10!,Q
R"
2%10!
,DF
"5%
2!,P
R"
5%2!.
!D
EF
"!
PQ
Rby
SS
S s
ince
cor
resp
ondi
ng s
ides
hav
e th
e sa
me
mea
sure
and
are
con
grue
nt.
2.D
(#7,
#3)
,E(#
4,#
1),F
(#2,
#5)
,P(2
,#2)
,Q(5
,#4)
,R(0
,#5)
DE
"%
13!,P
Q"
%13
,!
EF
"2%
5!,Q
R"
%26!
,DF
"%
29!,P
R"
%13!
.C
orre
spon
ding
sid
es a
re n
ot c
ongr
uent
,so
!D
EF
is n
ot c
ongr
uent
to
!P
QR
.
3.W
rite
a f
low
pro
of.
Giv
en:
R $S$
#T$
S$V
is t
he m
idpo
int
of R$
T$.
Pro
ve:
!R
SV
#!
TS
V
Pro
of:
Det
erm
ine
wh
ich
pos
tula
te c
an b
e u
sed
to
pro
ve t
hat
th
e tr
ian
gles
are
con
gru
ent.
If i
t is
not
pos
sibl
e to
pro
ve t
hat
th
ey a
re c
ongr
uen
t,w
rite
not
pos
sibl
e.
4.5.
6.
not
poss
ible
SA
S o
r S
SS
SS
S
7.IN
DIR
ECT
MEA
SUR
EMEN
TTo
mea
sure
the
wid
th o
f a
sink
hole
on
hi
s pr
oper
ty,H
arm
on m
arke
d of
f co
ngru
ent
tria
ngle
s as
sho
wn
in t
hedi
agra
m.H
ow d
oes
he k
now
tha
t th
e le
ngth
s A$
B$
and
AB
are
equa
l?S
ince
"A
CB
and
"A
#CB
#ar
e ve
rtic
al a
ngle
s,th
ey a
re
cong
ruen
t.In
the
fig
ure,
A!C!
"A!
#!C!an
d B!
C!"
B!#!C!
.So
!A
BC
"!
A#B
#Cby
SA
S.B
y C
PC
TC,t
he le
ngth
s A
#B#
and
AB
are
equa
l.A$
B$
AB
C
RS
" T
SGi
ven
SV "
SV
Refle
xive
Prop
erty
RV "
VT
Defin
ition
of m
idpo
int
V is
th
em
idp
oin
t o
f RT
. Gi
ven
!R
SV "
!TS
V
SSS
S
R V T
Pra
ctic
e (A
vera
ge)
Pro
ving
Con
grue
nce—
SS
S,S
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csP
rovi
ng C
ongr
uenc
e—S
SS
,SA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
5G
lenc
oe G
eom
etry
Lesson 4-4
Pre-
Act
ivit
yH
ow d
o la
nd
su
rvey
ors
use
con
gru
ent
tria
ngl
es?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
4 at
the
top
of
page
200
in y
our
text
book
.
Why
do
you
thin
k th
at la
nd s
urve
yors
wou
ld u
se c
ongr
uent
rig
ht t
rian
gles
rath
er t
han
othe
r co
ngru
ent
tria
ngle
s to
est
ablis
h pr
oper
ty b
ound
arie
s?S
ampl
e an
swer
:Lan
d is
usu
ally
div
ided
into
rec
tang
ular
lots
,so
the
ir b
ound
arie
s m
eet
at r
ight
ang
les.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
side
s of
!L
MN
for
whi
ch "
Lis
the
incl
uded
ang
le.
L!M!,L!
N!b.
Nam
e th
e si
des
of !
LM
Nfo
r w
hich
"N
is t
he in
clud
ed a
ngle
.
N!L!,
N!M!
c.N
ame
the
side
s of
!L
MN
for
whi
ch "
Mis
the
incl
uded
ang
le.
M!L!,
M!N!
2.D
eter
min
e w
heth
er y
ou h
ave
enou
gh in
form
atio
n to
pro
ve t
hat
the
two
tria
ngle
s in
eac
hfi
gure
are
con
grue
nt.I
f so
,wri
te a
con
grue
nce
stat
emen
t an
d na
me
the
cong
ruen
cepo
stul
ate
that
you
wou
ld u
se.I
f no
t,w
rite
not
pos
sibl
e.a.
b.
!A
BD
"!
CB
D;S
AS
not
poss
ible
c.E$
H$an
d D$
G$bi
sect
eac
h ot
her.
d.
!D
EF
"!
GH
F;S
AS
!R
SU
#!
TSU
;SS
S
Hel
pin
g Y
ou
Rem
emb
er
3.F
ind
thre
e w
ords
tha
t ex
plai
n w
hat
it m
eans
to
say
that
tw
o tr
iang
les
are
cong
ruen
t an
dth
at c
an h
elp
you
reca
ll th
e m
eani
ng o
f th
e SS
S Po
stul
ate.
Sam
ple
answ
er:C
ongr
uent
tri
angl
es a
re t
rian
gles
tha
t ar
e th
e sa
me
size
and
shap
e,an
d th
e S
SS
Pos
tula
te e
nsur
es t
hat
two
tria
ngle
s w
ith t
hree
corr
espo
ndin
g si
des
cong
ruen
t w
ill b
e th
e sa
me
size
and
sha
pe.
GE
F
HD
R T
SU
G
FD
E
CA
DB
L
N
M
©G
lenc
oe/M
cGra
w-H
ill20
6G
lenc
oe G
eom
etry
Con
grue
nt P
arts
of R
egul
ar P
olyg
onal
Reg
ions
Con
grue
nt f
igur
es a
re f
igur
es t
hat
have
exa
ctly
the
sam
e si
ze a
nd s
hape
.The
re a
re m
any
way
s to
div
ide
regu
lar
poly
gona
l reg
ions
into
con
grue
nt p
arts
.Thr
ee w
ays
to d
ivid
e an
equi
late
ral t
rian
gula
r re
gion
are
sho
wn.
You
can
veri
fy t
hat
the
part
s ar
e co
ngru
ent
bytr
acin
g on
e pa
rt,t
hen
rota
ting
,slid
ing,
or r
efle
ctin
g th
at p
art
on t
op o
f th
e ot
her
part
s.
1.D
ivid
e ea
ch s
quar
e in
to f
our
cong
ruen
t pa
rts.
Use
thr
ee
diff
eren
t w
ays.
Sam
ple
answ
ers
are
show
n.
2.D
ivid
e ea
ch p
enta
gon
into
fiv
e co
ngru
ent
part
s.U
se t
hree
di
ffer
ent
way
s.S
ampl
e an
swer
s ar
e sh
own.
3.D
ivid
e ea
ch h
exag
on in
to s
ix c
ongr
uent
par
ts.U
se t
hree
di
ffer
ent
way
s.S
ampl
e an
swer
s ar
e sh
own.
4.W
hat
hint
s m
ight
you
giv
e an
othe
r st
uden
t w
ho is
try
ing
to d
ivid
e fi
gure
s lik
e th
ose
into
con
grue
nt p
arts
?S
ee s
tude
nts’
wor
k.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-4
4-4
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Pro
ving
Con
grue
nce—
AS
A,A
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill20
7G
lenc
oe G
eom
etry
Lesson 4-5
ASA
Po
stu
late
The
Ang
le-S
ide-
Ang
le (
ASA
) Po
stul
ate
lets
you
sho
w t
hat
two
tria
ngle
sar
e co
ngru
ent.
AS
A P
ostu
late
If tw
o an
gles
and
the
incl
uded
sid
e of
one
tria
ngle
are
con
grue
nt to
two
angl
es
and
the
incl
uded
sid
e of
ano
ther
tria
ngle
, the
n th
e tr
iang
les
are
cong
ruen
t.
Fin
d t
he
mis
sin
g co
ngr
uen
t p
arts
so
that
th
e tr
ian
gles
can
be
pro
ved
con
gru
ent
by t
he
AS
A P
ostu
late
.Th
en w
rite
th
e tr
ian
gle
con
gru
ence
.
a.
Tw
o pa
irs
of c
orre
spon
ding
ang
les
are
cong
ruen
t,"
A#
"D
and
"C
#"
F.I
f th
ein
clud
ed s
ides
A$C$
and
D$F$
are
cong
ruen
t,th
en !
AB
C#
!D
EF
by t
he A
SA P
ostu
late
.
b.
"R
#"
Yan
d S$R$
# X$
Y$.I
f "
S#
"X
,the
n !
RS
T#
!Y
XW
by t
he A
SA P
ostu
late
.
Wh
at c
orre
spon
din
g p
arts
mu
st b
e co
ngr
uen
t in
ord
er t
o p
rove
th
at t
he
tria
ngl
esar
e co
ngr
uen
t by
th
e A
SA
Pos
tula
te?
Wri
te t
he
tria
ngl
e co
ngr
uen
ce s
tate
men
t.
1.2.
3.
D!C!
"B!
C!;
W!Y!
"W!
Y!;"
AB
E"
"C
BD
;!
CD
E"
!C
BA
"X
YW
""
ZYW
;!
AB
E"
!C
BD
!W
XY
"!
WZY
4.5.
6.
B!D!
"D!
B!;
S!T!"
V!T!;
"A
CB
""
E;
"A
DB
""
CB
D;
!R
ST
"!
UV
T!
AB
C"
!C
DE
!A
BD
"!
CD
B
AC
B
E
D
S
V
UR
T
D
AB
C
DC
EA
B
YW
X ZE
A
BD
C
RT
WY
SX
AC
B
DF
E
Exam
ple
Exam
ple
Exercis
esExercis
es
©G
lenc
oe/M
cGra
w-H
ill20
8G
lenc
oe G
eom
etry
AA
S Th
eore
mA
noth
er w
ay t
o sh
ow t
hat
two
tria
ngle
s ar
e co
ngru
ent
is t
he A
ngle
-A
ngle
-Sid
e (A
AS)
The
orem
.
AA
S T
heor
emIf
two
angl
es a
nd a
non
incl
uded
sid
e of
one
tria
ngle
are
con
grue
nt to
the
corr
espo
ndin
g tw
oan
gles
and
sid
e of
a s
econ
d tr
iang
le, t
hen
the
two
tria
ngle
s ar
e co
ngru
ent.
You
now
hav
e fi
ve w
ays
to s
how
tha
t tw
o tr
iang
les
are
cong
ruen
t.•
defi
niti
on o
f tr
iang
le c
ongr
uenc
e•
ASA
Pos
tula
te•
SSS
Post
ulat
e•
AA
S T
heor
em•
SAS
Post
ulat
e In t
he
dia
gram
,"B
CA
""
DC
A.W
hic
h s
ides
ar
e co
ngr
uen
t? W
hic
h a
dd
itio
nal
pai
r of
cor
resp
ond
ing
par
ts
nee
ds
to b
e co
ngr
uen
t fo
r th
e tr
ian
gles
to
be c
ongr
uen
t by
th
e A
AS
Pos
tula
te?
A $C$
# A$
C$by
the
Ref
lexi
ve P
rope
rty
of c
ongr
uenc
e.T
he c
ongr
uent
an
gles
can
not
be "
1 an
d "
2,be
caus
e A $
C$w
ould
be
the
incl
uded
sid
e.If
"B
#"
D,t
hen
!A
BC
#!
AD
Cby
the
AA
S T
heor
em.
In E
xerc
ises
1 a
nd
2,d
raw
an
d l
abel
!A
BC
and
!D
EF
.In
dic
ate
wh
ich
ad
dit
ion
alp
air
of c
orre
spon
din
g p
arts
nee
ds
to b
e co
ngr
uen
t fo
r th
e tr
ian
gles
to
beco
ngr
uen
t by
th
e A
AS
Th
eore
m.
1."
A#
"D
;"B
#"
E2.
BC
#E
F;"
A#
"D
If B!
C!"
E!F!(o
r if
A!C!
"D!
F!),
If "
C"
"F
(or
if "
B"
"E
),th
en !
AB
C"
!D
EF
by t
he
then
!A
BC
"!
DE
F by
the
AA
S T
heor
em.
AA
S T
heor
em.
3.W
rite
a f
low
pro
of.
Giv
en:"
S#
"U
;T $R$
bise
cts
"S
TU
.P
rove
:"S
RT
#"
UR
T
Give
n
Give
n
RT "
RT
Re
fl. P
rop.
of "
Def.o
f " b
isec
tor
TR b
isec
ts "
STU
.
!SR
T "
!U
RT
"ST
R "
"U
TR
AAS
"SR
T "
"U
RTCP
CTC
"S
" "
U
S
RT
U
BA
C
ED
F
CA
B
FD
E
D
C1 2
A
B
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Pro
ving
Con
grue
nce—
AS
A,A
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Pro
ving
Con
grue
nce—
AS
A,A
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill20
9G
lenc
oe G
eom
etry
Lesson 4-5
Wri
te a
flo
w p
roof
.
1.G
iven
:"
N#
"L
J $K$#
M$K$
Pro
ve:
!JK
N#
!M
KL
Pro
of:
2.G
iven
:A$
B$#
C$B$
"A
#"
CD $
B$bi
sect
s "
AB
C.
Pro
ve:
A $D$
#C$
D$
Pro
of:
3.W
rite
a p
arag
raph
pro
of.
Giv
en:
D$E$
|| F$G$
"E
#"
GP
rove
:!
DF
G#
!F
DE
Pro
of:S
ince
it is
giv
en t
hat
D!E!
|| F!G!
,it
follo
ws
that
"E
DF
""
GFD
,be
caus
e al
t.in
t.#
are
".I
t is
giv
en t
hat
"E
""
G.B
y th
e R
efle
xive
P
rope
rty,
D!F!
"F!D!
.So
!D
FG"
!FD
Eby
AA
S.
FG
DE
"A
" "
C
Give
nA
B "
CB
Give
nCP
CTC
AD
" C
D
DB
bis
ects
"A
BC
. Gi
ven
!A
BD
" !
CB
DAS
A
"A
BD
" "
CB
DDe
f. of
" b
isec
tor
AC
B D
"N
" "
LGi
ven
JK "
MK
Gi
ven
"JK
N "
"M
KL
Verti
cal #
are
".
!JK
N "
!M
KL
AAS
N
J
M
KL
©G
lenc
oe/M
cGra
w-H
ill21
0G
lenc
oe G
eom
etry
1.W
rite
a f
low
pro
of.
Giv
en:
Sis
the
mid
poin
t of
Q $T$
.Q $
R$|| T$
U$P
rove
:!
QS
R#
!T
SU
Sam
ple
proo
f:
2.W
rite
a p
arag
raph
pro
of.
Giv
en:
"D
#"
FG $
E$bi
sect
s "
DE
F.
Pro
ve:
D $G$
#F$G$
Pro
of:S
ince
it is
giv
en t
hat
G!E!
bise
cts
"D
EF,
"D
EG
""
FEG
by t
he
defin
ition
of
an a
ngle
bis
ecto
r.It
is g
iven
tha
t "
D"
"F.
By
the
Ref
lexi
ve P
rope
rty,
G!E!
"G!
E!.S
o !
DE
G"
!FE
Gby
AA
S.T
here
fore
D!
G!"
F!G!by
CP
CTC
.
AR
CH
ITEC
TUR
EF
or E
xerc
ises
3 a
nd
4,u
se t
he
foll
owin
g in
form
atio
n.
An
arch
itec
t us
ed t
he w
indo
w d
esig
n in
the
dia
gram
whe
n re
mod
elin
g an
art
stu
dio.
A $B$
and
C$B$
each
mea
sure
3 f
eet.
3.Su
ppos
e D
is t
he m
idpo
int
of A$
C$.D
eter
min
e w
heth
er !
AB
D#
!C
BD
.Ju
stif
y yo
ur a
nsw
er.
Sin
ce D
is t
he m
idpo
int
of A!
C!,A!
D!"
C!D!
by t
he d
efin
ition
of
mid
poin
t.A!
B!"
C!B!
by t
he d
efin
ition
of
cong
ruen
t se
gmen
ts.B
y th
e R
efle
xive
P
rope
rty,
B!D!
"B!
D!.S
o !
AB
D"
!C
BD
by S
SS
.
4.Su
ppos
e "
A#
"C
.Det
erm
ine
whe
ther
!A
BD
#!
CB
D.J
usti
fy y
our
answ
er.
We
are
give
n A!
B!"
C!B!
and
"A
""
C.B!
D!"
B!D!
by t
he R
efle
xive
P
rope
rty.
Sin
ce S
SA
can
not
be u
sed
to p
rove
tha
t tr
iang
les
are
cong
ruen
t,w
e ca
nnot
say
whe
ther
!A
BD
"!
CB
D.
DB
AC
D
G
F
E
"Q
" "
T
Give
n
QR
|| TU
Gi
ven
Def.o
f mid
poin
t
Alt.
Int.
# a
re "
.
QS
" T
S S
is t
he
mid
po
int
of
QT.
!Q
SR "
!TS
UAS
A
"Q
SR "
"TS
UVe
rtica
l # a
re "
.
UQ
S
RT
Pra
ctic
e (A
vera
ge)
Pro
ving
Con
grue
nce—
AS
A,A
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csP
rovi
ng C
ongr
uenc
e—A
SA
,AA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill21
1G
lenc
oe G
eom
etry
Lesson 4-5
Pre-
Act
ivit
yH
ow a
re c
ongr
uen
t tr
ian
gles
use
d i
n c
onst
ruct
ion
?R
ead
the
intr
oduc
tion
to
Les
son
4-5
at t
he t
op o
f pa
ge 2
07 in
you
r te
xtbo
ok.
Whi
ch o
f th
e tr
iang
les
in t
he p
hoto
grap
h in
you
r te
xtbo
ok a
ppea
r to
be
cong
ruen
t?S
ampl
e an
swer
:The
four
rig
ht tr
iang
les
are
cong
ruen
tto
eac
h ot
her.
The
two
obtu
se is
osce
les
tria
ngle
s ar
e co
ngru
ent
to e
ach
othe
r.R
ead
ing
th
e Le
sso
n1.
Exp
lain
in y
our
own
wor
ds t
he d
iffe
renc
e be
twee
n ho
w t
he A
SA P
ostu
late
and
the
AA
ST
heor
em a
re u
sed
to p
rove
tha
t tw
o tr
iang
les
are
cong
ruen
t.S
ampl
e an
swer
:In
AS
A,y
ou u
se t
wo
pair
s of
con
grue
nt a
ngle
s an
d th
ein
clud
edco
ngru
ent
side
s.In
AA
S,y
ou u
se t
wo
pair
s of
con
grue
nt a
ngle
san
d a
pair
of
noni
nclu
ded
cong
ruen
t si
des.
B,D
,E,G
,H2.
Whi
ch o
f the
follo
win
g co
ndit
ions
are
suf
ficie
nt t
o pr
ove
that
tw
o tr
iang
les
are
cong
ruen
t?A
.Tw
o si
des
of o
ne t
rian
gle
are
cong
ruen
t to
tw
o si
des
of t
he o
ther
tri
angl
e.B
.The
thr
ee s
ides
of
one
tria
ngle
s ar
e co
ngru
ent
to t
he t
hree
sid
es o
f th
e ot
her
tria
ngle
.C
.The
thr
ee a
ngle
s of
one
tri
angl
e ar
e co
ngru
ent
to t
he t
hree
ang
les
of t
he o
ther
tri
angl
e.D
.All
six
corr
espo
ndin
g pa
rts
of t
wo
tria
ngle
s ar
e co
ngru
ent.
E.T
wo
angl
es a
nd t
he in
clud
ed s
ide
of o
ne t
rian
gle
are
cong
ruen
t to
tw
o si
des
and
the
incl
uded
ang
le o
f th
e ot
her
tria
ngle
.F.
Tw
o si
des
and
a no
ninc
lude
d an
gle
of o
ne t
rian
gle
are
cong
ruen
t to
tw
o si
des
and
ano
ninc
lude
d an
gle
of t
he o
ther
tri
angl
e.G
.Tw
o an
gles
and
a n
onin
clud
ed s
ide
of o
ne t
rian
gle
are
cong
ruen
t to
tw
o an
gles
and
the
corr
espo
ndin
g no
ninc
lude
d si
de o
f th
e ot
her
tria
ngle
.H
.Tw
o si
des
and
the
incl
uded
ang
le o
f on
e tr
iang
le a
re c
ongr
uent
to
two
side
s an
d th
ein
clud
ed a
ngle
of
the
othe
r tr
iang
le.
I.T
wo
angl
es a
nd a
non
incl
uded
sid
e of
one
tri
angl
e ar
e co
ngru
ent
to t
wo
angl
es a
nd a
noni
nclu
ded
side
of
the
othe
r tr
iang
le.
3.D
eter
min
e w
heth
er y
ou h
ave
enou
gh in
form
atio
n to
pro
ve t
hat
the
two
tria
ngle
s in
eac
hfi
gure
are
con
grue
nt.I
f so
,wri
te a
con
grue
nce
stat
emen
t an
d na
me
the
cong
ruen
cepo
stul
ate
or t
heor
em t
hat
you
wou
ld u
se.I
f no
t,w
rite
not
pos
sibl
e.a.
!A
EB
"!
DE
C;A
AS
b.T
is t
he m
idpo
int
of R$
U$.
!R
ST
"!
UV
T;A
SA
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r m
athe
mat
ical
idea
s is
to
sum
mar
ize
them
in a
gen
eral
sta
tem
ent.
If y
ou w
ant
to p
rove
tri
angl
es c
ongr
uent
by
usin
g th
ree
pair
s of
cor
resp
ondi
ng p
arts
,w
hat
is a
goo
d w
ay t
o re
mem
ber
whi
ch c
ombi
nati
ons
of p
arts
will
wor
k?S
ampl
e an
swer
:At l
east
one
pai
r of
cor
resp
ondi
ng p
arts
mus
t be
side
s.If
you
use
two
pair
s of
sid
es a
nd o
ne p
air
of a
ngle
s,th
e an
gles
mus
t be
the
incl
uded
ang
les.
If yo
u us
e tw
o pa
irs
of a
ngle
s an
d on
e pa
ir o
f si
des,
then
the
sid
es m
ust
both
be
incl
uded
by
the
angl
es o
r m
ust
both
be
corr
espo
ndin
g no
ninc
lude
d si
des.
RS
T
U V
AD
CB
E
©G
lenc
oe/M
cGra
w-H
ill21
2G
lenc
oe G
eom
etry
Con
grue
nt T
rian
gles
in th
e C
oord
inat
e P
lane
If y
ou k
now
the
coo
rdin
ates
of
the
vert
ices
of
two
tria
ngle
s in
the
coo
rdin
ate
plan
e,yo
u ca
n of
ten
deci
de w
heth
er t
he t
wo
tria
ngle
s ar
e co
ngru
ent.
The
rem
ay b
e m
ore
than
one
way
to
do t
his.
1.C
onsi
der
!A
BD
and
!C
DB
who
se v
erti
ces
have
coo
rdin
ates
A(0
,0),
B(2
,5),
C(9
,5),
and
D(7
,0).
Bri
efly
des
crib
e ho
w y
ou c
an u
se w
hat
you
know
abo
ut c
ongr
uent
tri
angl
es a
nd t
he c
oord
inat
e pl
ane
to s
how
tha
t !
AB
D#
!C
DB
.You
may
wis
h to
mak
e a
sket
ch t
o he
lp g
et y
ou s
tart
ed.
Sam
ple
answ
er:S
how
tha
t th
e sl
opes
of
A !B !
and
C !D !
are
equa
l and
tha
t th
e sl
opes
of
A !D !
and
B !C !
are
equa
l.C
oncl
ude
that
A !B !
&C !
D !an
d B !
C ! &
A !D !
.Use
the
ang
le r
elat
ions
hips
for
para
llel l
ines
and
a t
rans
vers
al a
nd t
he f
act
that
B !D !
is a
com
-m
on s
ide
for
the
tria
ngle
s to
con
clud
e th
at
!A
BD
"!
CD
Bby
AS
A.
2.C
onsi
der
!P
QR
and
!K
LM
who
se v
erti
ces
are
the
follo
win
g po
ints
.
P(1
,2)
Q(3
,6)
R(6
,5)
K(#
2,1)
L(#
6,3)
M(#
5,6)
Bri
efly
des
crib
e ho
w y
ou c
an s
how
tha
t !
PQ
R#
!K
LM
.
Use
the
Dis
tanc
e Fo
rmul
a to
fin
d th
e le
ngth
s of
the
sid
es o
fbo
th t
rian
gles
.Con
clud
e th
at !
PQ
R"
!K
LMby
SS
S.
3.If
you
kno
w t
he c
oord
inat
es o
f al
l the
ver
tice
s of
tw
o tr
iang
les,
is it
al
way
spo
ssib
le t
o te
ll w
heth
er t
he t
rian
gles
are
con
grue
nt?
Exp
lain
.
Yes;
you
can
use
the
Dis
tanc
e Fo
rmul
a an
d S
SS
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-5
4-5
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Isos
cele
s Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
3G
lenc
oe G
eom
etry
Lesson 4-6
Pro
per
ties
of
Iso
scel
es T
rian
gle
sA
n is
osce
les
tria
ngl
eha
s tw
o co
ngru
ent
side
s.T
he a
ngle
for
med
by
thes
e si
des
is c
alle
d th
e ve
rtex
an
gle.
The
oth
er t
wo
angl
es a
re c
alle
dba
se a
ngl
es.Y
ou c
an p
rove
a t
heor
em a
nd it
s co
nver
se a
bout
isos
cele
s tr
iang
les.
•If
tw
o si
des
of a
tri
angl
e ar
e co
ngru
ent,
then
the
ang
les
oppo
site
th
ose
side
s ar
e co
ngru
ent.
(Iso
scel
es T
rian
gle
Th
eore
m)
•If
tw
o an
gles
of
a tr
iang
le a
re c
ongr
uent
,the
n th
e si
des
oppo
site
th
ose
angl
es a
re c
ongr
uent
.If
A$B$#
C$B$,
then
"A
#"
C.
If "
A#
"C
, the
n A$B$
#C$
B$.
A
BC
Fin
d x
.
BC
!B
A,s
o m
"A
!m
"C
.Is
os. T
riang
le T
heor
em
5x#
10 !
4x"
5S
ubst
itutio
n
x#
10 !
5S
ubtr
act 4
xfr
om e
ach
side
.
x!
15A
dd 1
0 to
eac
h si
de.
B
AC( 4
x $
5) !
( 5x
% 1
0)!
Fin
d x
.
m"
S!
m"
T,s
oS
R!
TR
.C
onve
rse
of Is
os. !
Thm
.
3x#
13 !
2xS
ubst
itutio
n
3x!
2x"
13A
dd 1
3 to
eac
h si
de.
x!
13S
ubtr
act 2
xfr
om e
ach
side
.
RT
S 3x %
13
2x
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exercis
esExercis
es
Fin
d x
.
1.35
2.12
3.15
4.12
5.20
6.36
7.W
rite
a t
wo-
colu
mn
proo
f.G
iven
:"1
#"
2P
rove
:A $B$
#C$
B$
Stat
emen
tsR
easo
ns
1."
1 "
"2
1.G
iven
2."
2 "
"3
2.Ve
rtic
al a
ngle
s ar
e co
ngru
ent.
3."
1 "
"3
3.Tr
ansi
tive
Pro
pert
y of
"4.
A!B!
"C!
B!4.
If tw
o an
gles
of
a tr
iang
le a
re "
,the
n th
e si
des
oppo
site
the
ang
les
are
".
B
AC
D
E
13
2
RS
T 3x!
x!D
BG
L3x
!
30!
D TQP
K
( 6x
$ 6
) !2x
!
W
YZ
3x!
S
V
T3x
% 6
2x $
6R
P
Q2x
!40
!
©G
lenc
oe/M
cGra
w-H
ill21
4G
lenc
oe G
eom
etry
Pro
per
ties
of
Equ
ilate
ral T
rian
gle
sA
n eq
uil
ater
al t
rian
gle
has
thre
e co
ngru
ent
side
s.T
he I
sosc
eles
Tri
angl
e T
heor
em c
an b
e us
ed t
o pr
ove
two
prop
erti
es o
f eq
uila
tera
ltr
iang
les.
1.A
tria
ngle
is e
quila
tera
l if a
nd o
nly
if it
is e
quia
ngul
ar.
2.E
ach
angl
e of
an
equi
late
ral t
riang
le m
easu
res
60°.
Pro
ve t
hat
if
a li
ne
is p
aral
lel
to o
ne
sid
e of
an
equ
ilat
eral
tri
angl
e,th
en i
t fo
rms
anot
her
equ
ilat
eral
tr
ian
gle.
Pro
of:
Stat
emen
tsR
easo
ns
1.!
AB
Cis
equ
ilate
ral;
P$Q$|| B$
C$.
1.G
iven
2.m
"A
!m
"B
!m
"C
!60
2.E
ach
"of
an
equi
late
ral !
mea
sure
s 60
°.3.
"1
#"
B,"
2 #
"C
3.If
||lin
es,t
hen
corr
es."
s ar
e #
.4.
m"
1 !
60,m
"2
!60
4.Su
bsti
tuti
on5.
!A
PQ
is e
quila
tera
l.5.
If a
!is
equ
iang
ular
,the
n it
is e
quila
tera
l.
Fin
d x
.
1.10
2.5
3.10
4.10
5.12
6.15
7.W
rite
a t
wo-
colu
mn
proo
f.G
iven
:!A
BC
is e
quila
tera
l;"
1 #
"2.
Pro
ve:"
AD
B#
"C
DB
Pro
of:
Stat
emen
tsR
easo
ns
1.!
AB
Cis
equ
ilate
ral.
1.G
iven
2.A!
B!"
C!B!
;"A
""
C2.
An
equi
late
ral !
has
"si
des
and
"an
gles
.3.
"1
""
23.
Giv
en4.
!A
BD
"!
CB
D4.
AS
A P
ostu
late
5."
AD
B"
"C
DB
5.C
PC
TC
A D C
B1 2
R O
HM
60!
4x!
X
ZY
4x %
4
3x $
860
!
PQ
LV
R60
!
4x40
L N M
K
!K
LM is
equ
ilate
ral.
3x!
G
JH
6x %
55 x
D
FE
6x!
A
B
PQ
C
12
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Isos
cele
s Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Isos
cele
s Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
5G
lenc
oe G
eom
etry
Lesson 4-6
Ref
er t
o th
e fi
gure
.
1.If
A$C$
#A$
D$,n
ame
two
cong
ruen
t an
gles
."
AC
D"
"C
DA
2.If
B$E$
#B$
C$,n
ame
two
cong
ruen
t an
gles
."
BE
C"
"B
CE
3.If
"E
BA
#"
EA
B,n
ame
two
cong
ruen
t se
gmen
ts.
E!B!"
E!A!
4.If
"C
ED
#"
CD
E,n
ame
two
cong
ruen
t se
gmen
ts.
C!E!
"C!
D!
!A
BF
is i
sosc
eles
,!C
DF
is e
quil
ater
al,a
nd
m"
AF
D"
150.
Fin
d e
ach
mea
sure
.
5.m
"C
FD
606.
m"
AF
B55
7.m
"A
BF
708.
m"
A55
In t
he
figu
re,P!
L!"
R!L!
and
L!R!
"B!
R!.
9.If
m"
RL
P!
100,
find
m"
BR
L.
20
10.I
f m
"L
PR
!34
,fin
d m
"B
.68
11.W
rite
a t
wo-
colu
mn
proo
f.
Giv
en:
C$D$
#C $
G$D$
E$#
G$F$
Pro
ve:
C$E$
#C$
F$
Pro
of:
Sta
tem
ents
Rea
sons
1.C!
D!"
C!G!
1.G
iven
2."
D"
"G
2.If
2 si
des
of a
!ar
e "
,the
n th
e #
oppo
site
thos
e si
des
are
".
3.D!
E!"
G!F!
3.G
iven
4.!
CD
E"
!C
GF
4.S
AS
5.C!
E!"
C!F!
5.C
PC
TC
D E F G
CRP
BL
D
C F
B
35!
AE
D
C
B
AE
©G
lenc
oe/M
cGra
w-H
ill21
6G
lenc
oe G
eom
etry
Ref
er t
o th
e fi
gure
.
1.If
R$V$
#R$
T$,n
ame
two
cong
ruen
t an
gles
."
RTV
""
RV
T
2.If
R$S$
#S$V$
,nam
e tw
o co
ngru
ent
angl
es.
"S
VR
""
SR
V
3.If
"S
RT
#"
ST
R,n
ame
two
cong
ruen
t se
gmen
ts.
S!T!"
S!R!
4.If
"S
TV
#"
SV
T,n
ame
two
cong
ruen
t se
gmen
ts.
S!T!"
S!V!
Tri
angl
es G
HM
and
HJ
Mar
e is
osce
les,
wit
h G!
H!"
M!H!
and
H!J!
"M!
J!.T
rian
gle
KL
Mis
equ
ilat
eral
,an
d m
"H
MK
"50
.F
ind
eac
h m
easu
re.
5.m
"K
ML
606.
m"
HM
G70
7.m
"G
HM
40
8.If
m"
HJM
!14
5,fi
nd m
"M
HJ.
17.5
9.If
m"
G!
67,f
ind
m"
GH
M.
46
10.W
rite
a t
wo-
colu
mn
proo
f.
Giv
en:
D$E$
|| B$C$
"1
#"
2P
rove
:A $
B$#
A$C$
Pro
of:
Sta
tem
ents
Rea
sons
1.D!
E!|| B!
C!1.
Giv
en
2."
1 "
"4
2.C
orr.
#ar
e "
."
2 "
"3
3."
1 "
"2
3.G
iven
4."
3 "
"4
4.C
ongr
uenc
e of
#is
tra
nsiti
ve.
5.A!
B!"
A!C!
5.If
2 #
of a
!ar
e "
,the
n th
e si
des
oppo
site
th
ose
#ar
e "
.
11.S
PORT
SA
pen
nant
for
the
spo
rts
team
s at
Lin
coln
Hig
h Sc
hool
is in
the
sha
pe o
f an
isos
cele
s tr
iang
le.I
f th
e m
easu
re
of t
he v
erte
x an
gle
is 1
8,fi
nd t
he m
easu
re o
f ea
ch b
ase
angl
e.81
,81
Linc
oln
Haw
ks
E DBC
A12
3 4
GMLK
J
H
UR
TV
S
Pra
ctic
e (A
vera
ge)
Isos
cele
s Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csIs
osce
les
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
7G
lenc
oe G
eom
etry
Lesson 4-6
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n a
rt?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
6 at
the
top
of
page
216
in y
our
text
book
.
•W
hy d
o yo
u th
ink
that
isos
cele
s an
d eq
uila
tera
l tri
angl
es a
re u
sed
mor
eof
ten
than
sca
lene
tri
angl
es in
art
?S
ampl
e an
swer
:The
irsy
mm
etry
is p
leas
ing
to t
he e
ye.
•W
hy m
ight
isos
cele
s ri
ght
tria
ngle
s be
use
d in
art
?S
ampl
e an
swer
:Tw
o co
ngru
ent
isos
cele
s ri
ght
tria
ngle
s ca
n be
pla
ced
toge
ther
to
form
a s
quar
e.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.W
hat
kind
of
tria
ngle
is !
QR
S?
isos
cele
sb.
Nam
e th
e le
gs o
f !Q
RS
.Q!
S!,R!
S!c.
Nam
e th
e ba
se o
f !Q
RS
.Q!
R!d.
Nam
e th
e ve
rtex
ang
le o
f !Q
RS
."
Se.
Nam
e th
e ba
se a
ngle
s of
!Q
RS
."
Q,"
R
2.D
eter
min
e w
heth
er e
ach
stat
emen
t is
alw
ays,
som
etim
es,o
r ne
ver
true
.
a.If
a t
rian
gle
has
thre
e co
ngru
ent
side
s,th
en it
has
thr
ee c
ongr
uent
ang
les.
alw
ays
b.If
a t
rian
gle
is is
osce
les,
then
it is
equ
ilate
ral.
som
etim
esc.
If a
rig
ht t
rian
gle
is is
osce
les,
then
it is
equ
ilate
ral.
neve
rd.
The
larg
est
angl
e of
an
isos
cele
s tr
iang
le is
obt
use.
som
etim
ese.
If a
rig
ht t
rian
gle
has
a 45
°an
gle,
then
it is
isos
cele
s.al
way
sf.
If a
n is
osce
les
tria
ngle
has
thr
ee a
cute
ang
les,
then
it is
equ
ilate
ral.
som
etim
esg.
The
ver
tex
angl
e of
an
isos
cele
s tr
iang
le is
the
larg
est
angl
e of
the
tri
angl
e.so
met
imes
3.G
ive
the
mea
sure
s of
the
thr
ee a
ngle
s of
eac
h tr
iang
le.
a.an
equ
ilate
ral t
rian
gle
60,6
0,60
b.an
isos
cele
s ri
ght
tria
ngle
45,4
5,90
c.an
isos
cele
s tr
iang
le in
whi
ch t
he m
easu
re o
f th
e ve
rtex
ang
le is
70
70,5
5,55
d.an
isos
cele
s tr
iang
le in
whi
ch t
he m
easu
re o
f a
base
ang
le is
70
70,7
0,40
e.an
isos
cele
s tr
iang
le in
whi
ch t
he m
easu
re o
f th
e ve
rtex
ang
le is
tw
ice
the
mea
sure
of
one
of t
he b
ase
angl
es90
,45,
45
Hel
pin
g Y
ou
Rem
emb
er4.
If a
the
orem
and
its
conv
erse
are
bot
h tr
ue,y
ou c
an o
ften
rem
embe
r th
em m
ost
easi
ly b
yco
mbi
ning
the
m in
to a
n “i
f-an
d-on
ly-if
”sta
tem
ent.
Wri
te s
uch
a st
atem
ent
for
the
Isos
cele
sT
rian
gle
The
orem
and
its
conv
erse
.S
ampl
e an
swer
:Tw
o si
des
of a
tri
angl
e ar
eco
ngru
ent
if an
d on
ly if
the
ang
les
oppo
site
tho
se s
ides
are
con
grue
nt.
R Q
S
©G
lenc
oe/M
cGra
w-H
ill21
8G
lenc
oe G
eom
etry
Tria
ngle
Cha
lleng
esSo
me
prob
lem
s in
clud
e di
agra
ms.
If y
ou a
re n
ot s
ure
how
to
solv
e th
epr
oble
m,b
egin
by
usin
g th
e gi
ven
info
rmat
ion.
Fin
d th
e m
easu
res
of a
s m
any
angl
es a
s yo
u ca
n,w
riti
ng e
ach
mea
sure
on
the
diag
ram
.Thi
s m
ay g
ive
you
mor
e cl
ues
to t
he s
olut
ion.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-6
4-6
1.G
iven
:B
E!
BF
, "B
FG
# "
BE
F #
"B
ED
, m"
BF
E !
82 a
ndA
BF
Gan
d B
CD
Eea
ch h
ave
oppo
site
sid
es p
aral
lel a
ndco
ngru
ent.
Fin
d m
"A
BC
.14
8
3.G
iven
:m
"U
ZY
!90
,m"
ZW
X!
45,
!Y
ZU
#!
VW
X,U
VX
Yis
asq
uare
(al
l sid
es c
ongr
uent
,all
angl
es r
ight
ang
les)
.F
ind
m"
WZ
Y.
45
2.G
iven
:A
C!
AD
,and
A $B $
#B $
D $,
m"
DA
C!
44 a
ndC $
E $bi
sect
s "
AC
D.
Fin
d m
"D
EC
.78
4.G
iven
:m
"N
!12
0,J $N $
#M $
N $,
!JN
M#
!K
LM
.F
ind
m"
JKM
.15
J
K
L
MN
A
DC
BE
UV
W
XY
Z
A
GD
FE
CB
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Tria
ngle
s an
d C
oord
inat
e P
roof
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill21
9G
lenc
oe G
eom
etry
Lesson 4-7
Posi
tio
n a
nd
Lab
el T
rian
gle
sA
coo
rdin
ate
proo
f us
es p
oint
s,di
stan
ces,
and
slop
es t
opr
ove
geom
etri
c pr
oper
ties
.The
fir
st s
tep
in w
riti
ng a
coo
rdin
ate
proo
f is
to
plac
e a
figu
re o
nth
e co
ordi
nate
pla
ne a
nd la
bel t
he v
erti
ces.
Use
the
fol
low
ing
guid
elin
es.
1.U
se th
e or
igin
as
a ve
rtex
or
cent
er o
f the
figu
re.
2.P
lace
at l
east
one
sid
e of
the
poly
gon
on a
n ax
is.
3.K
eep
the
figur
e in
the
first
qua
dran
t if p
ossi
ble.
4.U
se c
oord
inat
es th
at m
ake
the
com
puta
tions
as
sim
ple
as p
ossi
ble.
Pos
itio
n a
n e
quil
ater
al t
rian
gle
on t
he
coor
din
ate
pla
ne
so t
hat
its
sid
es a
re a
un
its
lon
g an
d
one
sid
e is
on
th
e p
osit
ive
x-ax
is.
Star
t w
ith
R(0
,0).
If R
Tis
a,t
hen
anot
her
vert
ex is
T(a
,0).
For
vert
ex S
,the
x-c
oord
inat
e is
%a 2% .U
se b
for
the
y-co
ordi
nate
,
so t
he v
erte
x is
S%%a 2% ,
b &.
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
1.2.
3.
C(p
,q)
T(2
a,2a
)E
(%2g
,0);
F(0
,b)
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
4.is
osce
les
tria
ngle
5.
isos
cele
s ri
ght
!D
EF
6.eq
uila
tera
l tri
angl
e !
EQ
I!
RS
T w
ith
base
R $S$
wit
h le
gs e
unit
s lo
ngw
ith
vert
ex Q
(0,a
) an
d4a
unit
s lo
ngsi
des
2bun
its
long x
y
I( b, 0
)E
( –b,
0)
Q( 0
, a)
x
y
E( e
, 0)
F( e
, e)
D( 0
, 0)
x
yT
( 2a,
b)
R( 0
, 0)
S( 4
a, 0
)
Sam
ple
answ
ers
are
give
n.
x
y
G( 2
g, 0
)
F( ?
, b)
E( ?
, ?)
x
y
S( 2
a, 0
)
T( ?
, ?)
R( 0
, 0)
x
y
B( 2
p, 0
)
C( ?
, q)
A( 0
, 0)
x
y
T( a
, 0)
R( 0
, 0)S
#a – 2, b$
Exercis
esExercis
es
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill22
0G
lenc
oe G
eom
etry
Wri
te C
oo
rdin
ate
Pro
ofs
Coo
rdin
ate
proo
fs c
an b
e us
ed t
o pr
ove
theo
rem
s an
d to
veri
fy p
rope
rtie
s.M
any
coor
dina
te p
roof
s us
e th
e D
ista
nce
Form
ula,
Slop
e Fo
rmul
a,or
Mid
poin
t T
heor
em.
Pro
ve t
hat
a s
egm
ent
from
th
e ve
rtex
an
gle
of a
n i
sosc
eles
tri
angl
e to
th
e m
idp
oin
t of
th
e ba
se
is p
erp
end
icu
lar
to t
he
base
.F
irst
,pos
itio
n an
d la
bel a
n is
osce
les
tria
ngle
on
the
coor
dina
te
plan
e.O
ne w
ay is
to
use
T(a
,0),
R(#
a,0)
,and
S(0
,c).
The
n U
(0,0
) is
the
mid
poin
t of
R $T$
.
Giv
en:I
sosc
eles
!R
ST
;Uis
the
mid
poin
t of
bas
e R $
T$.
Pro
ve:S $
U$⊥
R$T$
Pro
of:
Uis
the
mid
poin
t of
R $T$
so t
he c
oord
inat
es o
f Uar
e %%#
a 2"a
%,%
0" 2
0%
&!(0
,0).
Thu
s S$U$
lies
on
the
y-ax
is,a
nd !
RS
Tw
as p
lace
d so
R$T$
lies
on t
he x
-axi
s.T
he a
xes
are
perp
endi
cula
r,so
S $U$
⊥R$
T$.
Pro
ve t
hat
th
e se
gmen
ts j
oin
ing
the
mid
poi
nts
of
the
sid
es o
f a
righ
t tr
ian
gle
form
a ri
ght
tria
ngl
e.
Sam
ple
answ
er:P
ositi
on a
nd la
bel r
ight
!A
BC
with
the
coo
rdin
ates
A
(0,0
),B
(0,2
b),a
nd C
(2a,
0).
The
mid
poin
t P
of B
Cis
#&0$ 2
2a &,&
2b2$
0&
$"(a
,b).
The
mid
poin
t Q
of A
Cis
#&0$ 2
2a &,&
0$ 2
0&
$"(a
,0).
The
mid
poin
t R
of A
Bis
#&0$ 2
0&
,&0
$ 22b &
$"(0
,b).
The
slop
e of
R!P!
is &b a
% %b 0
&"
&0 a&"
0,so
the
seg
men
t is
hor
izon
tal.
The
slop
e of
P!Q!
is &b a
%%a0
&"
&b 0& ,w
hich
is u
ndef
ined
,so
the
segm
ent
is v
ertic
al.
"R
PQ
is a
rig
ht a
ngle
bec
ause
any
hor
izon
tal l
ine
is p
erpe
ndic
ular
to
any
vert
ical
line
.!P
RQ
has
a ri
ght
angl
e,so
!P
RQ
is a
rig
ht t
rian
gle.
x
y
C( 2
a, 0
)
B( 0
, 2b)
P Q
R
A( 0
, 0)
x
y
T( a
, 0)
U( 0
, 0)
R( –
a, 0
)
S( 0
, c)
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Tria
ngle
s an
d C
oord
inat
e P
roof
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-7
4-7
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 4-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Tria
ngle
s an
d C
oord
inat
e P
roof
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill22
1G
lenc
oe G
eom
etry
Lesson 4-7
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1.ri
ght
!F
GH
wit
h le
gs
2.is
osce
les
!K
LP
wit
h 3.
isos
cele
s !
AN
Dw
ith
aun
its
and
bun
its
base
K$P$
6bun
its
long
base
A$D$
5alo
ng
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
4.5.
6.
A(0
,2a)
Z(b
,c)
M(0
,c)
7.8.
9.
Q(4
a,0)
R#&7 2& b
,c$
T(0
,b)
10.W
rite
a c
oord
inat
e pr
oof
to p
rove
tha
t in
an
isos
cele
s ri
ght
tria
ngle
,the
seg
men
t fr
omth
e ve
rtex
of
the
righ
t an
gle
to t
he m
idpo
int
of t
he h
ypot
enus
e is
per
pend
icul
ar t
o th
ehy
pote
nuse
.
Giv
en:
isos
cele
s ri
ght
!A
BC
wit
h "
AB
Cth
e ri
ght
angl
e an
d M
the
mid
poin
t of
A $C$
Pro
ve:
B$M$
⊥A$
C$
Pro
of:
The
Mid
poin
t Fo
rmul
a sh
ows
that
the
coo
rdin
ates
of
Mar
e #&0
$ 22a &
,&2a
2$0
&$o
r (a
,a).
The
slop
e of
A!C!
is
&2 0a %%
20 a&
"%
1.Th
e sl
ope
of B!
M!is
&a a% %
0 0&
"1.
The
prod
uct
of t
he s
lope
s is
%1,
so B!
M!⊥
A!C!
.
x
y
C( 2
a, 0
)
A( 0
, 2a)
M
B( 0
, 0)
x
y
U( a
, 0)
T( ?
, ?)
S( –
a, 0
)x
y
P( 7
b, 0
)
R( ?
, ?)
N( 0
, 0)
x
y
Q( ?
, ?)
R( 2
a, b
)
P( 0
, 0)
x
y
N( 3
b, 0
)
M( ?
, ?)
O( 0
, 0)
x
y
Y( 2
b, 0
)
Z( ?
, ?)
X( 0
, 0)
x
y
B( 2
a, 0
)
A( 0
, ?)
C( 0
, 0)
x
yN
#5 – 2a, b
$
A( 0
, 0)
D( 5
a, 0
)x
yL(
3b, c
)
K( 0
, 0)
P( 6
b, 0
)x
y F( 0
, a)
G( 0
, 0)
H( b
, 0)
Sam
ple
answ
ers
are
give
n.
©G
lenc
oe/M
cGra
w-H
ill22
2G
lenc
oe G
eom
etry
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1.eq
uila
tera
l !S
WY
wit
h 2.
isos
cele
s !
BL
Pw
ith
3.is
osce
les
righ
t !
DG
Jsi
des
%1 4% alo
ngba
se B$
L$3b
uni
ts lo
ngw
ith
hypo
tenu
se D$
J$an
dle
gs 2
aun
its
long
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
4.5.
6.
S#&1 6& b
,c$
C(3
a,0)
,E(0
,c)
M(0
,c),
N(%
2b,0
)
NEI
GH
BO
RH
OO
DS
For
Exe
rcis
es 7
an
d 8
,use
th
e fo
llow
ing
info
rmat
ion
.K
arin
a liv
es 6
mile
s ea
st a
nd 4
mile
s no
rth
of h
er h
igh
scho
ol.A
fter
sch
ool s
he w
orks
par
tti
me
at t
he m
all i
n a
mus
ic s
tore
.The
mal
l is
2 m
iles
wes
t an
d 3
mile
s no
rth
of t
he s
choo
l.
7.W
rite
a c
oord
inat
e pr
oof
to p
rove
tha
t K
arin
a’s
high
sch
ool,
her
hom
e,an
d th
e m
all a
reat
the
ver
tice
s of
a r
ight
tri
angl
e.
Giv
en:
!S
KM
Pro
ve:
!S
KM
is a
rig
ht t
rian
gle.
Pro
of:
Slo
pe o
f S
K"
&4 6% %
0 0&
or &2 3&
Slo
pe o
f S
M"
& %3 2% %0 0
&or
%&3 2&
Sin
ce t
he s
lope
of
S!M!is
the
neg
ativ
e re
cipr
ocal
of
the
slop
e of
S!K!
,S!M!
⊥S!K!
.The
refo
re,!
SK
Mis
rig
ht t
rian
gle.
8.F
ind
the
dist
ance
bet
wee
n th
e m
all a
nd K
arin
a’s
hom
e.
KM
"%
(%2
%!
6)2
$!
(3 %
!4)
2!
"%
64 $
!1!
"%
65!or
' 8
.1 m
iles
x
y S( 0
, 0)
K( 6
, 4)
M( –
2, 3
)
x
y
P( 2
b, 0
)
M( 0
, ?)
N( ?
, 0)
x
y
C( ?
, 0)
E( 0
, ?)
B( –
3a, 0
)x
yS
( ?, ?
)
J(0,
0)
R#1 – 3b,
0$
x
y D( 0
, 2a)
G( 0
, 0)
J(2a
, 0)
x
yP
#3 – 2b, c
$
B( 0
, 0)
L(3b
, 0)
x
yY
#1 – 8a, b
$
W#1 – 4a,
0$
S( 0
, 0)
Sam
ple
answ
ers
are
give
n.
Pra
ctic
e (A
vera
ge)
Tria
ngle
s an
d C
oord
inat
e P
roof
NA
ME
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4-7
4-7
Answers (Lesson 4-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csTr
iang
les
and
Coo
rdin
ate
Pro
of
NA
ME
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ATE
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RIO
D__
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4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill22
3G
lenc
oe G
eom
etry
Lesson 4-7
Pre-
Act
ivit
yH
ow c
an t
he
coor
din
ate
pla
ne
be u
sefu
l in
pro
ofs?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 4-
7 at
the
top
of
page
222
in y
our
text
book
.
Fro
m t
he c
oord
inat
es o
f A,B
,and
Cin
the
dra
win
g in
you
r te
xtbo
ok,w
hat
do y
ou k
now
abo
ut !
AB
C?
Sam
ple
answ
er:!
AB
Cis
isos
cele
sw
ith "
Cas
the
ver
tex
angl
e.
Rea
din
g t
he
Less
on
1.F
ind
the
mis
sing
coo
rdin
ates
of
each
tri
angl
e.a.
b.
R(0
,b),
S(0
,0),
T#a,
&b 2& $D
(0,0
),E
(0,a
),F
(a,a
)
2.R
efer
to
the
figu
re.
a.F
ind
the
slop
e of
S$R$
and
the
slop
e of
S$T$
.1;
%1
b.F
ind
the
prod
uct
of t
he s
lope
s of
S$R$
and
S$T$.W
hat
does
thi
s te
ll yo
u ab
out
S $R$an
d S$T$
?%
1;S!R!
⊥S!T!
c.W
hat
does
you
r an
swer
fro
m p
art
b te
ll yo
u ab
out
!R
ST
?S
ampl
e an
swer
:!R
ST
is a
rig
ht t
rian
gle
with
"S
as t
he r
ight
ang
le.
d.F
ind
SR
and
ST
.Wha
t do
es t
his
tell
you
abou
t S$R$
and
S$T$?
SR
"%
2a2
!or
a%
2!;S
T"
%2a
2!
or a
%2!;
S!R!"
S!T!e.
Wha
t do
es y
our
answ
er f
rom
par
t d
tell
you
abou
t !
RS
T?
Sam
ple
answ
er:!
RS
Tis
isos
cele
s w
ith "
RS
Tas
the
ver
tex
angl
e.f.
Com
bine
you
r an
swer
s fr
om p
arts
c a
nd e
to
desc
ribe
!R
ST
as c
ompl
etel
y as
pos
sibl
e.S
ampl
e an
swer
:!R
ST
is a
n is
osce
les
righ
t tr
iang
le."
RS
Tis
the
rig
htan
gle
and
is a
lso
the
vert
ex a
ngle
.g.
Fin
d m
"S
RT
and
m"
ST
R.
45;4
5h
.F
ind
m"
OS
Ran
d m
"O
ST
.45
;45
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stud
ents
fin
d it
eas
ier
to r
emem
ber
mat
hem
atic
al f
orm
ulas
if t
hey
can
put
them
into
wor
ds in
a c
ompa
ct w
ay.H
ow c
an y
ou u
se t
his
appr
oach
to
rem
embe
r th
e sl
ope
and
mid
poin
t fo
rmul
as e
asily
?S
ampl
e an
swer
:Slo
pe F
orm
ula:
chan
ge in
yov
er c
hang
e in
x;
Mid
poin
t Fo
rmul
a:av
erag
e of
x-c
oord
inat
es,a
vera
ge o
f y-
coor
dina
tes
x
y S( 0
, a)
R( –
a, 0
)T
( a, 0
)O
( 0, 0
)
x
y
F( ?
, ?)
E( ?
, a)
D( ?
, ?)
x
y
T( a
, ?)
R( ?
, b)
S( ?
, ?)
©G
lenc
oe/M
cGra
w-H
ill22
4G
lenc
oe G
eom
etry
How
Man
y Tr
iang
les?
Eac
h pu
zzle
bel
ow c
onta
ins
man
y tr
iang
les.
Cou
nt t
hem
car
eful
ly.
Som
e tr
iang
les
over
lap
othe
r tr
iang
les.
How
man
y tr
ian
gles
are
th
ere
in e
ach
fig
ure
?
1.8
2.40
3.35
4.5
5.13
6.27
How
man
y tr
ian
gles
can
you
for
m b
y jo
inin
g p
oin
ts o
n e
ach
cir
cle?
L
ist
the
vert
ices
of
each
tri
angl
e.
7.8.
4;A
BC
,AB
D,A
CD
,BC
D10
;EFG
,EFH
,EFI
,EG
H,E
HI,
FGH
,FG
I,FH
I,E
GI,
GH
I
8.9.
20;J
KL,
JKM
,JK
N,J
KO
,JLM
,JL
N,J
LO,J
MN
,JM
O,J
NO
,KLM
,K
LN,K
LO,K
MN
,KM
O,K
NO
,LM
N,L
MO
,LN
O,M
NO
35;
PQ
R,P
QS
,PQ
T,P
QU
,PQ
V,
PR
S,P
RT,
PR
U,P
RV
,PS
T,P
SU
,P
SV
,PTU
,PTV
,PU
V,Q
RS
,QR
T,Q
RU
,QR
V,Q
ST,
QS
U,Q
SV
,QTU
,Q
TV,Q
UV
,RS
T,R
SU
,RS
V,R
TU,
RTV
,RU
V,S
TU,S
TV,S
UV,
TUV
QR
P
U
S
TV
JK
O
L
MN
EF
I
GH
B
C
DA
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
4-7
4-7
Answers (Lesson 4-7)