Unit 4a Triangle Relationships Page 1 of 23 Geometry Unit 4a - Notes This unit is broken into two parts, 4a & 4b. A test should be given following each part. Triangle Relationships Triangle ∆ - a figure formed by three segments joining three noncollinear points, called vertices. The triangle symbol is . Syllabus Objective 4.1 - The student will classify triangles by sides and/or angles. Classification of Triangles by Sides Equilateral triangle - a triangle with three congruent sides. Isosceles triangle - a triangle with at least two congruent sides Scalene triangle - a triangle with no congruent sides. Classification of Triangles by Angles Acute triangle - a triangle with three acute angles. Right triangle - a triangle with one right angle. Obtuse triangle - a triangle with one obtuse angle. Equiangular triangle – a triangle with three congruent angles. Adjacent sides of a triangle - two sides sharing a common vertex. Adjacent Sides Common Vertex Hypotenuse of a right triangle - the side opposite the right angle.
Transcript
Unit 4a Triangle Relationships Page 1 of 23
Geometry Unit 4a - Notes
This unit is broken into two parts, 4a & 4b. A test should be given following each part. Triangle Relationships
Triangle
∆ - a figure formed by three segments joining three noncollinear points, called
vertices. The triangle symbol is . Syllabus Objective 4.1 - The student will classify triangles by sides and/or angles.
Classification of Triangles by Sides Equilateral triangle
- a triangle with three congruent sides.
Isosceles triangle
- a triangle with at least two congruent sides
Scalene triangle
- a triangle with no congruent sides.
Classification of Triangles by Angles Acute triangle
- a triangle with three acute angles.
Right triangle
- a triangle with one right angle.
Obtuse triangle
- a triangle with one obtuse angle.
Equiangular triangle
– a triangle with three congruent angles.
Adjacent sides of a triangle
- two sides sharing a common vertex.
Adjacent Sides Common Vertex Hypotenuse of a right triangle
- the side opposite the right angle.
Unit 4a Triangle Relationships Page 2 of 23
3
2
1
Syllabus Objective 4.2 - The student will solve problems applying the triangle sum and exterior angle theorems. Ask the entire class to draw a triangle on a piece of paper, then have each person cut out their triangle. Label the angles 1, 2, and 3 as shown. Tear each angle from the triangle and then place them side by side along a straight line. The three angles seem to fill in or form straight line. Because by definition a straight angle is 180º, the experiment might lead students to believe the sum of the interior angles of a triangle is 180º.
13
2
While that’s not a proof, it does provide students with some valuable insights. The fact is, it turns out to be true.
BaseBase
Vertexl
Base
Vertex
LegLeg
Isosceles Triangle
Acutel
Acute
Rightl
LegHypotenuse
Leg
Right Triangle
Unit 4a Triangle Relationships Page 3 of 23
Triangle Sum Theorem:{
The sum of the measures of the interior angles of a triangle is 180º. ∆ Sum Th.}
Given: DEF∆ Prove: 1 2 3 180m m m∠ + ∠ + ∠ = °
The most important part of any proof is the ability to use geometry already learned. In the case, if only the three angles of the triangle were considered, the proof would go nowhere. Using parallel lines, from the previous unit, construct RS
parallel to DE and label the angles formed. Parallel lines being cut by a
transversal create angle pairs that will help complete the proof.
Statements Reasons 1) DEF∆ 1) Given 2) Draw RS DE
2) Construction
3) 4 & DFS∠ ∠ are supp. 3) L.P. Post. 4) 4 180m m DFS∠ + ∠ = ° 4) Def of Supp. 5) 2 5m DFS m m∠ = ∠ + ∠ 5) ∠Add. Post. 6) 4 2 5 180m m m∠ + ∠ + ∠ = ° 6) Substitution
7) 4,
3 51≅∠
∠ ≅ ∠
∠ 7) lines cut by trans., alt. int. 's∠ ≅
8) 1 2 3 180m m m∠ + ∠ + ∠ = ° 8) Substitution A theorem that follows directly from this theorem is one about the relationship
between the exterior angle of a triangle and the nonadjacent angles inside the triangle. Exterior angle – when the sides of a triangle are extended, the angles that are adjacent to the interior angles.
F S R
E D
5 4
3
2
1
Unit 4a Triangle Relationships Page 4 of 23
3
60°
50°
Exterior Angle Theorem:∠
The exterior angle of a triangle is equal to the sum of the two remote interior angles. {Ext. Th.} Given: ABC∆ Prove: 1m m A m C∠ = ∠ + ∠
Statements Reasons 1) 2 180m A m C m∠ + ∠ + ∠ = ° 1) ∆ Sum Th. 2) 1 & 2∠ ∠ are supp. 's∠ 2) L.P. Post. 3) 1 2 180m m∠ + ∠ = ° 3) Def. of supp. 4) 2 1 2m A m C m m m∠ + ∠ + ∠ = ∠ + ∠ 4) Substitution 5) 1m A m C m∠ + ∠ = ∠ 5) Subtr. Prop. of Equality
Example:
Find the measure of ∠3.
Since the sum of the two angles given is 110˚, ∠3 must be 70˚.
Corollary to Triangle Sum Theorem
: The acute angles of a right triangle are complementary.
Examples: a)
Find the value of x.
The sum of the interior angles is 180˚.
30˚ + (2x + 10)˚ + (3x)˚ = 180˚ (combine like terms) 5x + 40 = 180 (subtract 40) 5x = 140 (divide by 5) x = 28˚
(2x + 10) °
30 °
3x °
C
B A 1 2
Unit 4a Triangle Relationships Page 5 of 23
b)
Exterior angle is equal to the sum of the two remote interior angles.
(2x + 10)˚ = x˚ + 60˚ (subtract x) x + 10 = 60 (subtract 10) x = 50˚
Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent. {3rd ∠ Th.} Syllabus Objective 4.3 - The student will analyze the relationships between congruent figures.
The word congruent is used to describe objects that have the same size and shape. Things that are traced are considered congruent. After being traced, the objects could be flipped or rotated but they will still maintain their congruence.
Congruence
Congruent objects have corresponding or matching sides and angles. In order for objects to be congruent, all their corresponding sides and all their corresponding angles must be congruent to one another. Writing congruence statements: when stating that two polygons are congruent the corresponding angles must be written in order.
x◦
60 °
(2x + 10) °
Stop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be congruent octagons.
C B
A
F E
D
ABC DEF≅∆ ∆
Unit 4a Triangle Relationships Page 6 of 23
Triangles
Mathematically, all the sides and angles of one triangle must be congruent to the corresponding sides and angles of another triangle.
In other words, it must be shown that angles A, B, and C are congruent (≅ ) to angles D, E and F, and , ,AB BC and AC shown are ≅ to , ,DE EF and DF respectively. ALL six relationships must be shown.
Some triangle congruence theorems:
Reflexive Property of Congruent Triangles – Every triangle is congruent to itself. Symmetric Property of Congruent Triangles – If ,ABC DEF∆ ≅ ∆ then .DEF ABC∆ ≅ ∆ Transitive Property of Congruent Triangles - If ABC DEF∆ ≅ ∆ and ,DEF JKL∆ ≅ ∆ then
.ABC JKL∆ ≅ ∆ Syllabus Objective 4.4 - The student will justify congruence using corresponding parts of congruent triangles. Rather than showing all the angles and all the sides of one triangle are congruent to all the sides and all the angles of another triangle (6 relationships), students are now able to determine congruence by just using the 3 sets of corresponding sides. A shortcut! That leads to the Side, Side, Side congruence postulate.
C B
A
F E
D
If given three sticks of length 10”, 8”, and 7” and asked to glue the ends together to make triangles, students would find that something interesting happens. The triangles would all stack on top of each other, they would coincide. Because they are congruent!
To determine if two triangles are congruent, they must have the same size and shape. They must fit on top of each other, they must coincide.
Unit 4a Triangle Relationships Page 7 of 23
Side-Side-Side Congruence Postulate: If three sides of one triangle are congruent, respectively, to three sides of another triangle, then the triangles are congruent. {SSS}
Using similar demonstrations, two more congruence postulates can be arrived at. The Side, Angle, Side postulate is abbreviated SAS Postulate. Side-Angle-Side Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent. {SAS}
A third postulate is the Angle, Side, Angle postulate. Angle-Side-Angle Congruence Postulate: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. {ASA}
C B
A
F E
D
C B
A
F E
D
C B
A
F E
D
Combining this information with previous information, students will be able to determine if triangles are congruent.
Unit 4a Triangle Relationships Page 8 of 23
Syllabus Objective 4.6 - The student will prove that two triangles are congruent.
Proofs: Congruent ∆'s To prove triangles are congruent, use the SSS, SAS and ASA congruence postulates. Also, review other theorems that will lead students to more information. Angle-Angle-Side Congruence Theorem
: If two angles and the non included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. {AAS}
Example:
Write a two-column proof. Draw and label the two triangles.
Given: ,,
DC FA
AB DE
≅ ∠
∠ ≅∠
≅
∠
Prove: ABC DEF≅∆ ∆
Statements Reasons
1) ,,
DC FA
AB DE
≅ ∠
∠ ≅∠
≅
∠
1) Given
2) EB ≅∠∠ 2) 3rd ∠ Th 3) ABC DEF≅∆ ∆ 3) ASA
B A
C
E D
F
Unit 4a Triangle Relationships Page 9 of 23
4 ways of proving triangles congruent: SSS, SAS, ASA, and AAS.
When writing proofs:
Given: AC BD , bisectsAB CD
Prove: ACX BDX≅∆ ∆
B
D X C
A
**Students should be reminded to mark their diagrams with given information. They should add any additional information that is found through their investigation.
First, label congruences in the diagram using previous knowledge. Second, label any “unspoken” information that must be included (like vertical angles or shared sides). Next, try to use one of the four methods (SSS, SAS, ASA, and AAS) of proving triangles congruent. Finally, write those relationships in the body of proof.
Visualization is very important and helpful in completing proofs!
Right Triangles While the congruence postulates and theorems apply for all triangles, we have postulates and theorems that apply specifically for right triangles. HL Theorem: If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.
B
D X C
A
In order to prove these triangles congruent, the definition of a bisector and the subsequent mathematical relationship was necessary. Even though vertical angles were not part of the given information, they are “unspoken” information and can be seen in the diagram. Therefore, the theorem that all vertical angles are congruent can be used.
The other congruence theorems for right triangles can be seen as special cases of the other triangle congruence postulates and theorems.
Unit 4a Triangle Relationships Page 11 of 23
LL Theorem: If two legs of one right triangle are congruent to two legs of another right triangle, the triangles are congruent. HA Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of another right triangle, the triangles are congruent. LA Theorem: If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, the triangles are congruent.
As marked in this first set, ASA.
As marked in this second set, AAS.
Proofs: CPCTC When two triangles are congruent, each part of one triangle is congruent to the corresponding part of the other triangle. That’s referred to as Corresponding Parts of Congruent Triangles are Congruent, thus CPCTC. One way to determine if two line segments or two angles are congruent is by showing they are the corresponding parts of two congruent triangles.
Special case of SAS and can be considered redundant.
AAS and can be considered redundant.
If drawn and labeled, the diagram of the LA Congruence Theorem would show that it is derived from either ASA or AAS depending on which corresponding set of legs or angles are congruent. Once
again this theorem can be considered redundant.
Unit 4a Triangle Relationships Page 12 of 23
1. Identify two triangles in which the segments or angles are the corresponding parts. 2. Prove the triangles are congruent. 3. State the two parts are congruent, supporting the statement with the reason;
“corresponding parts of congruent triangles are congruent”. Proving two segments congruent:
Given: and bisect each otherAB CD
Prove: AD BC≅ The strategy to prove these segments are congruent is to first show the triangles are congruent. Fill in the diagram showing the relationships based upon the information given and the other relationships that exist using previously learned definitions, theorems, and postulates. Using the definition of bisector, it is determined that AP PB≅ and DP PC≅ . Also, notice a pair of vertical angles. The diagram has been marked to show these relationships. Even though the vertical angles are marked in the diagram, the statement must be written in the proof.
B C
P
D A
B C
P
D A
Unit 4a Triangle Relationships Page 13 of 23
Statements Reasons
1) and bisect each otherAB CD 1) Given
2) ,AP PB CP PD≅ ≅ 2) Def. of bisector
3) CPBAPD ∠∠ ≅ 3) V.A. Th. 4) APD BPC∆ ≅∆ 4) SAS
5) AD BC≅ 5) CPCTC
Filling in the body of the proof is easier after marking the congruences in the diagram. The strategy to show angles or segments are congruent is to first show the triangles are congruent, then use CPCTC.
Given: ,AC BCAX BX
≅
≅
Prove: 21 ∠∠ ≅
In this case: CX CX≅ .
AXC BXC∆ ≅∆
If the triangles are congruent, then all the remaining corresponding parts of the triangles are congruent by CPCTC. That means 21 ∠∠ ≅ .
2 1
C
X B A
2 1
C
X B A
**Mark the picture with the parts that are congruent based on what’s given. **Then mark the relationships based upon knowledge of geometry.
The diagram shows three sides of one triangle are congruent to three corresponding sides of another triangle; therefore the triangles are congruent
Unit 4a Triangle Relationships Page 14 of 23
Syllabus Objective 4.7 - The student will prove and use the properties of isosceles and/or equilateral triangles. Base Angle Theorem
: If two sides of a triangle are congruent, then the angles opposite them are congruent.
Corollary to the Base Angle Theorem
: If a triangle is equilateral, then it is equiangular.
Converse of the Base Angle Theorem
: If two angles of a triangle are congruent, then the sides opposite them are congruent.
Given:,B
ABCA ≅∠
∆∠
Prove: AC BC≅
Statements Reasons
1) Draw ∠ bisector CX 1) Construction 2) 21 ∠∠ ≅ 2) Def. of ∠ bisector 3) BA ∠∠ ≅ 3) Given
4) CX CX≅ 4) Reflexive Prop.
5) CAX CBX∆ ≅∆ 5) AAS
6) AC BC≅ 6) CPCTC
Corollary to the Converse of the Base Angle Theorem: If a triangle is equiangular, then it is equilateral.
The idea of using CPCTC after proving triangles congruent by SSS, SAS, ASA, and ASA will allow students to find many more relationships in geometry.
C
B A
2 1
C
X B A
Unit 4a Triangle Relationships Page 15 of 23
The Problem: **Isosceles Triangle**
In the Isosceles triangle shown, CA = CB. From A, a AP
has been drawn to meet the opposite side BC at right angles (altitude from A).
Show that 12
PAB ACB∠ = ∠ or 12
a b= .
In Solution:
, 90 .APB a c∆ = ° − In ( ), 180 2 2 90 .ABC b c c∆ = ° − = ° −
Hence 1 .22
or a bb a ==
The Problem: **Two Triangles**
Two equilateral triangles, of lengths 10 and 7 respectively, are drawn, with their bases touching and in line.
710
Unit 4a Triangle Relationships Page 16 of 23
Lines are drawn to connect the tops of each to the furthest corner of the other. Is one line longer than the other? Can you prove it?
This can be solved by using the Pythagorean Theorem, but is much easier than that. Solution:
Rotate ABD∆ 60° clockwise about B. It’s congruent to EBC∆ ! Therefore, AD = EC.
Syllabus Objective 4.5 - The student will solve problems related to congruent triangles using algebraic techniques.
Example: Corresponding angles: 4x = 52 → x = 13.
Given the two congruent triangles, find the values of x and y.
The other acute angle would be (90 – 52)° or 38°. So 6y + 2 = 38 → 6y = 36 → y = 6.
Syllabus Objective 4.8 - The student will classify triangles using coordinate geometry. (May need supplemental material)
4x°
(6y + 2)°
52°
Students must be able to match corresponding parts of congruent triangles and solve for unknown values and measures. They may have to determine that the triangles are congruent before they can perform the calculations.
E
D
CBA60°
710
Unit 4a Triangle Relationships Page 17 of 23
Z ( a , ?)
Y (?, ?)
X (?, ?) O
Coordinate proofs
– use figures in the coordinate plane and algebra to prove geometric concepts.
Some helpful hints: (Whenever possible) Use the origin as a vertex or center of the triangle. Place at least one side of a triangle on an axis. Keep the triangle within the first quadrant. Use coordinates that make computations as simple as possible.
Isosceles triangle can be placed as so: Right triangles as so: {Increments have been removed to show how coordinate graphs can be illustrated. Assign variable lengths; a, b, c, etc…}
Example:
Name the missing coordinates of isosceles triangle XYZ.
Vertex X is positioned at the origin; its coordinates are (0, 0). Vertex Z is on the x-axis, so its y-coordinate is 0. The coordinates of vertex Z are (a, 0). ∆XYZ is isosceles, so the x-coordinate of Y is
located halfway between 0 and a, or at2a . We
cannot write the y-coordinate in terms of a, so call it
b. The coordinates of point Y are
,2a b .
It is helpful to discuss convenient placement of figures in the coordinate plane. When coordinates are not given, care should be taken to place figures in positions that benefit the work to be done.
Unit 4a Triangle Relationships Page 18 of 23
Example:
Write a coordinate proof to show that a line segment joining the midpoints of two sides of a triangle is parallel to the third side.
Place a vertex at the origin and label it A. Use coordinates that are multiples of 2 because the Midpoint Formula involves dividing the sum of the coordinates by 2.
Given: ∆ABC , S is the midpoint of AC , T is the midpoint of BC
Prove: ST AB Proof:
By the Midpoint Formula, the coordinates of S are + +
2 0 2 0,2 2
b c or ( ),b c
and the coordinates of T are + +
2 2 0 2,2 2
a b c or ( )+ ,a b c .
By the Slope Formula, the slope of ST is −+ −c c
a b b or 0 and the slope of AB
is −−
0 02a a
or 0. Since ST and AB have the same slope, ST AB .
What is the reason that the statement in Step 4 is true?
A. side-angle-side B. angle-side-angle C. Opposite sides of a parallelogram are
congruent. D. Corresponding angles of congruent triangles
are congruent.
The statements for a proof are given below. Given: Parallelogram ABCD BAX DCY∠ ≅ ∠ Prove: BX DY≅
Proof:
STATEMENTS REASONS 1. Parallelogram ABCD,
BAX DCY∠ ≅ ∠ 1. Given
2. B D∠ ≅ ∠ 2. 3. AB CD≅ 3. 4. ABX CDY∆ ≅ ∆ 4. 5. BX DY≅ 5.
What reason makes the statement in Step 4 true?
A. Side-angle-side congruence theorem. B. Angle-side-angle congruence theorem. C. Opposite sides of a parallelogram are
congruent. D. Corresponding parts of congruent triangles
are congruent.
C
D
A
B
X
Y
C
D
A
B
X
Y
Unit 4a Triangle Relationships Page 22 of 23
26. The statements for a proof are given below. Given: AB FD≅ B D∠ ≅ ∠ A F∠ ≅ ∠ Prove: BC DE≅
Proof: STATEMENTS REASONS 1. AB FD≅ 1. Given 2. B D∠ ≅ ∠ 2. Given 3. A F∠ ≅ ∠ 3. Given 4. ABC FDE∆ ≅ ∆ 4. ______
5. BC DE≅ 5. Corresponding Parts of Congruent Triangles are Congruent
What is the missing reason that would complete this proof?
A. side-side-side B. side-angle-side C. angle-side-angle D. angle-angle-side
The statements for a proof are given below. Given: AB FD≅ A F∠ ≅ ∠ C E∠ ≅ ∠ Prove: BC DE≅
Proof: STATEMENTS REASONS 1. AB FD≅ 1. Given 2. A F∠ ≅ ∠ 2. Given 3. C E∠ ≅ ∠ 3. Given 4. ABC FDE∆ ≅ ∆ 4. ______ 5. BC DE≅ 5. ______
What reason makes the statement in Step 5 true?
A. Angle-angle-side congruence theorem. B. Angle-side-angle congruence theorem. C. Definition of congruent segments. D. Corresponding parts of congruent triangles
are congruent.
27. Given that DEF LMN∆ ≅ ∆ , ( )2 15m D x∠ = + ° , ( )3 2m L x ∠ = − ° ,
and 4( 17)DF x= − , what is LN? A. 16 B. 21 C. 57 D. 67
Given that DEF LMN∆ ≅ ∆ , ( )75m D x∠ = + ° , ( )3 15m L x∠ = + ° , and
2 26DF x= − , what is LN? A. 28 B. 34 C. 101 D. 105
28. In the isosceles triangle below, 137m H∠ = ° .
What is the measure of F∠ ?
A. 21.5° B. 26.5° C. 43° D. 53°
In the isosceles triangle below, 124m H∠ = ° .
What is the measure of F∠ ?
A. 28° B. 56° C. 124° D. 180°
H
G
124°
F
F
H
G
137°
C
D
A
B
F
E
C
D
A
B
F
E
Unit 4a Triangle Relationships Page 23 of 23
Sample SAT Question(s):
Taken from College Board online practice problems.
1. In isosceles triangle ABC above, AM and CM are the angle bisectors of angle BAC and angle BCA. What is the measure of angle AMC?
(A) 110° (B) 115° (C) 120° (D) 125° (E) 130°
2. In the figure above, point B lies on side AC . If 55 60x< < , what is one possible value of y?
sdGrid-In
3. If XYX∆ is equilateral, what is the value of r s t u+ + + ?
