Day In Class Homework Completed Day 1 Classifying Triangles D1 HW – pg. 5 Day 2 Angles of Triangles D2 HW – pg. 11 Day 3 Angles of Triangles and Congruent Triangles Skills Practice 4-2 Day 4 Isosceles and Equilateral Triangles D4 HW – pg. 17 Day 5 Coordinate Triangle Proofs D5 HW – pg. 22 (#1) Day 6 Coordinate Triangle Proofs D6 HW – pg. 22 (#2) Day 7 Congruent Triangles D7 HW – pg. 26 Day 8 SSS and SAS Worksheet in Packet Day 9 ASA and AAS Review Packet Day 10 Review Study Day 11 Test Good Luck! 1
Transcript
Day In Class Homework Completed
Day 1 Classifying Triangles D1 HW – pg. 5
Day 2 Angles of Triangles D2 HW – pg. 11
Day 3 Angles of Triangles and Congruent Triangles Skills Practice 4-2
The graphic organizer describes the relationships of interior and exterior angles in a triangle. Use the word bank to correctly identify each.
Third Angle Theorem Right Triangles Exterior Angle TheoremTriangle Sum Theorem Equiangular Triangles
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Third Angle Theorem – If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.
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1
6
4
5
Isosceles and Equilateral Triangles
In an isosceles triangle,________ sides are equal, therefore ________ angles are equal.
Vertex Angle-
Base-
Base Angles-
Legs-
Theorem ExampleIsosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite the sides are congruent.
If RT RS, then _____ _____.
Converse of Isosceles Triangle Theorem (ITBA)If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If N M, then _____ _____.
Remember Equilateral Triangles: all sides are ________ to each other. RT RS ST all angles are ________. R S T
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A
B C
T S
R
N M
L
T S
R
1. Find the mB and mC if mA = 53.
2. Find the measure of each angle.
a. b.
mC = ________ mP = ________
c. d.
mH = ________ mA = ________ mB = ________
3. Given the isosceles triangle, find mY and mZ.
4. BCA is an equilateral triangle. Find the value of x and y.
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A C
B
X
Z Y
8840
B
A C
78
PR
Q
86
B
A C78
H
G
J
8x
6x + 18
5. Find the value of x.
6. Find the measure of the vertex angle if each base angle is 80.
7. Find the measure of each base angle if the vertex angle is 76.
8. What is the measure of degrees in each acute angle of an isosceles right triangle?
9. The degree measure of the vertex angle A of an isosceles triangle is 110. Find the other angles.
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B
A C
4y - 532x
2x
W
X Y
88
62 4x - 2
10. In the accompanying diagram, triangle ABC and triangle ABD are isosceles triangles with the mCAB = 50 and mBDA = 55. If AC = AB and AB = BD, what is the mCBD?
11.The measure of the vertex angle of an isosceles triangle is 15 more than each base angle. Find the number of degrees in each angle.
12.The measure of the base angle of an isosceles triangle is seven times the measure of the vertex angle. Find the number of degrees in each angle.
13. In isosceles triangle ABC, the measure of the vertex angle C is 30 more than each base angle. Find the number of degrees in each angle.
14.The measure of the exterior angle to a base angle of an isosceles triangle is 115. What is the measure of the vertex angle of the triangle?
15. In a triangle, the measure of the second angle is 30 more than the measure of the first angle. The measure of the third angle is 45 more than the first angle. Find the number of degrees in each angle of the triangle. What type of triangle is it?
16. In triangle ABC, mA = x, mB = x + 10, and the measure of an exterior angle at C is 70. Find the value of x.
Day 4 HW
1. Find each measure.16
A
D
B
C
a. mBAC b. mSRT
c. CB d. TR
2. Find the value of x.
a. b. c.
3. Find each measure.
a. mCAD
b. mACD
c. mACB
d. mABC
Triangles and Coordinates Proofs
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C
BA 60
T
R
S50
R
P
T
443 55
55
C
B
A
6x - 9 2x + 11
3x + 6
24x2 + 5x
D C B
A
92
Coordinate proofs use figures in the coordinate plane and algebra to prove geometric concepts.
To Do Coordinate Geometry Proofs:1. Graph the figure. 2. Use one or a combination of the following formulas:
Distance formula to show that __________ are __________.
Slope formula to show that __________ are __________ or have
____________________.
Midpoint formula to show that __________ have the __________ midpoint.
3 Show ALL work.
4. Write a statement(s) to explain why it is that figure to finish the proof.
Properties of an Isosceles Triangle:1. An isosceles triangle has __________ sides __________.
2. An isosceles triangle has __________ angles __________.
Show two sides have the same distance. That means sides will have the same
__________.
To prove that 2 sides have the same length, use the __________
formula. You will do the distance formula _____ times.
Statement(s):
1. The coordinates of triangle ABC are A(5, 4), B(8, 1), and C(2, 1). Prove that ABC is an isosceles triangle. Hint: Show that AB AC.
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2. The coordinates of triangle ABC are A(3, 1), B(1, -1), and C(5, -1). Prove that ABC is an isosceles triangle.
Properties of a Right Triangle:
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1. A right triangle has _____ right __________.
Show the triangle has a right angle. That means the slopes of the legs of the triangle
will have ____________________________.
To prove that there is a right angle, use the __________ formula. You will do
the slope formula _____ times. The slopes will have ____________________.
Statement(s):
1. The coordinates of triangle ABC are A(-1, 1), B(-4, 1), and C(-1, 3). Prove that ABC is a right triangle. Hint: Show that AB and AC form a right angle.
2. The coordinates of triangle ABC are A(-1, -1), B(2, -3), and C(-1, -3). Prove that ABC is a right triangle.
Properties of an Isosceles Right Triangle:20
1. An isosceles right triangle has _____ right __________.
2. An isosceles right triangle has _____ equal __________.
Show two sides have the same distance and there is a right angle.
To prove a triangle is isosceles, use the __________ formula. You will do the
distance formula _____ times. The lengths will have ____________________.
AND To prove that there is a right angle, use the __________ formula. You will do
the slope formula _____ times. The slopes will have ____________________.
Statement(s):
1. The coordinates of triangle RST are R(0, 1), S(4, 5), and T(4, 1). Prove that RST is an isosceles right triangle.
Days 5 and Day 6 Homework
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1. The coordinates of triangle ABC are A(2, 5), B(5, 2), and C(-1, 2). Prove that ABC is an isosceles triangle.
2. The coordinates of triangle ABC are A(-1, 1), B(1, -2), and C(-1, -2). Prove that ABC is a right triangle.
Congruent Triangles
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Remember: congruent means ________ shape, ________ size. The symbol for congruent is _____.
Triangles are congruent if they have the same size and shape. Their corresponding parts, the angles and sides that are in the same position, are congruent.
To identify corresponding parts of congruent triangles, look at the order of the vertices in the congruence statement such as ABC JLK.
1. Given: XYZ NPQ. Identify the congruent corresponding parts.
a. Q _____ b. _____ c. P _____
d. X _____ e. _____ f. _____
g. Write a congruence statement.
2. Use the given information to find the measures of the angles.
Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive.
Reflexive Property of Triangle Congruence:Ex: A A , AB ABABC _____
Symmetric Property of Triangle Congruence:Ex: AB BAIf ABC EFG, then _______________
Transitive Property of Triangle Congruence:Ex: If 1 2 and 2 3, then 1 3If ABC EFG and EFG JKL , then _______________
Day 7 HW
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1. Show that the triangles are congruent by identifying all congruent parts. Then write a congruence statement.
a. R _____ b. _____ c. S _____
d. T _____ e. _____ f. _____
g. Congruence Statement _____________________
2. Find x and y.
3. Find each measure.
a. m1
b. m2
c. m3
4. Which is a factor of x2 + 19x – 42?
a. x + 14 b. x + 2 c. x – 2 d. x – 14
5. Find the distance between points (5, 7) and (-2, 3).
Proving Triangles Congruent
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T
R J
L
S K
y 402x
3
7415
2
1
To prove triangles are congruent, you must find _____ corresponding parts that match up.
Triangle Congruence Theorems: SSS Congruence
When proving triangles congruent with SSS, you must find _____ sides.
You write in Justifications: _______________.
SAS CongruenceWhen proving triangles congruent with SAS, you must find _____ sides and _____ angle. You must have _____ included ___________.The word included means __________.
You write in Justifications: _______________.
ASA Congruence
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SSS SAS
How are they alike? How are they different?
When proving triangles congruent with ASA, you must find _____ side and _____ angles. You must have _____ included ___________.
You write in Justifications: _______________.
AAS CongruenceWhen proving triangles congruent with AAS, you must find _____ side and _____ angles. The side is not included.
You write in Justifications: _______________.
From the given information, what theorem (SSS, SAS, ASA or AAS) would prove the triangles congruent.
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ASA AAS
How are they alike? How are they different?
1. A C, AD DC 2. HE FG, EFH FHG, EHF GFH
3. U W, UV WV, UX WX 4. UV WV, UX WX
5. Y B, YA BA 6. DEG FEG, DE EF
7. HJK MLK, JH ML 8. A D, AB DE, F C
Write a two-column proof.
1. Given: AB DB C is the midpoint of AD.
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B
A CD
F
E
H
G
V
W
U
X V
W
U
X
C B
Y
A
Z
G
FDE
H
K
J M
L
A
F
D
CB E
Prove: ABC DBC
2. Given: AB ll CD CAB ACD
Prove: ACD ABC
3. Given: RS TU, RT US Prove: RST UTS
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Statements Reasons
A B
CD
Statements Reasons
A
B
C D
Statements Reasons
S
T U
R
4. Given: S V T is the midpoint of SV
Prove: RST UTV
5. Given: CD bisects AB AC BC
Prove: ADC BDC
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Statements ReasonsU
V
T
S
R
C
BA D
Statements Reasons
6. Given: T is the midpoint of PQ PQ bisects RS RQ SP
