Download Packet #2 - White Plains Public Schools

Triangle
Congruence
Packet #2
Name ____________________________ Teacher _____________
1
Table of Contents
Day 1 : SWBAT: Prove Triangles Congruent using SSS and SAS Theorems
Pages 3 - 8
HW: Pages 9 - 11
Day 2: SWBAT: Prove Triangles Congruent using ASA and AAS Theorems
Pages 12 - 16
HW: pages 17 - 19
Day 3: SWBAT: Prove Triangles Congruent using HL Theorem
Pages 20-26
HW: pages 27 - 28
Day 4: SWBAT: Proving Overlapping and Isosceles Triangles Congruent
Pages 29 - 34
HW: pages 35 - 37
Day 5: SWBAT: Use CPCTC to Prove Segment Relationships of Triangles
Pages 38 - 43
HW: pages 44 - 46
Day 6&7: SWBAT: Review of Proving Triangles Congruent
Pages 47- 53
Day 8: Test
2
Day 1 – Proving Triangles Congruent by SSS and SAS
Warm - Up
Which of the following condition(s) does not prove that two triangles are congruent?
1) SSS ≅ SSS
2) SSA ≅ SSA
3) SAS ≅ SAS
4) ASA ≅ ASA
5) AAA ≅ AAA
3
Model Problem #1
Given: ̅̅̅̅ ̅̅̅̅
D is the midpoint of ̅̅̅̅
Prove:
Given
Given
_____  ______
_____  ______
_________  _________
2)
Given
_____  ______
_____  ______
_____  ______
____________
 _________
____________
 _________
4
3)
Given
d
___
 _________
_____  ______
Given
_____  ______
You Try It!
4)
5
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
Given
_____  ______
_____  ______
You Try It!
6
Challenge
SUMMARY
Example 1: Proving Triangles Congruent
7
SUMMARY
Exit Ticket
8
Homework
1.
2.
9
3.
4) Given: ̅̅̅̅
̅̅̅̅, ⃗⃗⃗⃗⃗
Prove:
10
5)
̅̅̅̅̅
̅̅̅̅̅
6.
11
Day 2 – Proving Triangles Congruent by ASA and AAS
Warm-Up
What additional information would you need to prove these
triangles congruent by ASA?
_____  ______
Model Problem #1
Prove: (a) YWZ  YWX
(b) ̅̅̅̅
̅̅̅̅
Given
_____  ______
___
 _________
_____  ______
_____  ______
12
You Try It!
13
14
Challenge
SUMMARY of the ASA Theorem
15
SUMMARY of the AAS Theorem
Exit Ticket
16
Day 2 - AAS/ASA Proofs Homework
1.
2.
17
3.
4.
Prove:
 P
18
5. Given: ̅̅̅̅ ̅̅̅̅, T is the midpoint of ̅̅̅̅.
Prove: ̅̅̅̅ ̅̅̅̅
\
5.
6.
19
Day 3 – HL
20
Example 1:
21
You Try It!
Example 2:
Given: ̅̅̅̅̅
Prove:
̅̅̅̅̅ ̅̅̅̅

̅̅̅̅
̅̅̅̅̅  ̅̅̅̅
22
Theorem: All radii of a circle are congruent!
You Try It!

23
You Try It!
Exit Ticket
24
Challenge
25
Summary
HL Theorem
Proofs Involving Circles
26
Homework
1.
2. Given: ̅̅̅̅
Prove:
̅̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅

27
3.
4.
28
Day 4 – Overlapping and Isosceles Triangles
Warm – Up
29
3.
Draw the triangles separately here:
4.

Draw the triangles separately here:
30
Using Isosceles Triangles in Proofs
When we studied triangles in earlier chapters, we learned about the
properties of the isosceles triangles. We can use these ideas to help us
when constructing proofs.
Example 5:
31
You Try It!
Given: XYZ is isosceles with base ̅̅̅̅.
̅̅̅̅̅
̅̅̅̅
Prove:

Example 6:
Given: ̅̅̅̅
Prove:
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
is an Isosceles Triangle
32
Challenge
Exit Ticket
33
SUMMARY of Overlapping Triangles
\\\\\\\\\\\\\
Summary of Isosceles Triangles
34
Homework
1.
2.
3.
35
4.
5. Given: ∆PMR is isosceles with base ̅̅̅̅̅
Prove: ∆PSM
36
6. Given: ̅̅̅̅
̅̅̅̅̅
Prove:
̅̅̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
̅̅̅̅
7.
37
Day 5 – CPCTC and Beyond
Warm - Up
Complete the proof below by filling in the reasons.
Statements
Reasons
38
Example 1:
ADC
BDC
CPCTC
Example 2:
Prove: B is the midpoint of ̅̅̅̅
39
Ex 3:
Ex 4:
Given: ̅̅̅̅
Prove: ̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅
40
Altitudes
An altitude of a triangle is a line segment drawn from any vertex of a triangle,
perpendicular to the opposite side of the triangle.
Ex 5: Given:
̅̅̅̅
Prove: ̅̅̅̅
̅̅̅̅
̅̅̅
̅̅̅ and
41
Proving Perpendicular Segments
Theorem: Two angles are congruent and supplementary  each angle is a right angle
Ex 5:
Prove: ̅̅̅̅
̅̅̅̅̅
Challenge
42
SUMMARY
Exit Ticket
Which Theorem below would not be used in justifying the congruence below?
43
Homework
1.
2.
44
3. Given: ̅̅̅̅
Prove: ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ is an altitude to ̅̅̅̅
Prove: ̅̅̅̅ bisects
4. Given:
45
̅̅̅̅ is an altitude to ̅̅̅̅̅
5. Given:
Prove: ̅̅̅̅ bisects ̅̅̅̅̅
46
Day 6 – REVIEW
47
Concept Summary
Addition and Subtraction
Use ADDITION when:
a) you have a gap.
b) you need larger pieces.
Use SUBTRACTION when:
a) you have overlap.
b) you need smaller pieces.
CPCTC
Stands for: Corresponding Parts of Congruent Triangles are Congruent
Use when you are asked to prove SEGMENTS or ANGLES congruent.
Proofs involving Circles
Theorem: All radii are congruent!
Isosceles Triangle Theorem
Or
Overlapping Triangles
Draw triangles separately. Mark up both the together and separate diagrams.
Know how to draw conclusions from “Key” Vocabulary Words
1.
2.
3.
4.
5.
6.
7.
8.
Midpoint → 2  segments
Bisector → 2  segments
Angle bisector → 2  angles
Perpendicular ( ) → right angles → all right angles are 
Median → Midpoint → 2  segments
Altitude → Perpendicular ( ) → right angles → all right angles are 
Vertical Angles → angles are 
→ Alternating Interior Angles (A.I.A’s) 
48
Geometry Practice Proofs
1.
2.
3.
49
4.
5.
6.
50
̅̅̅̅
̅̅̅̅ bisects ̅̅̅̅̅
̅̅̅̅
51
̅̅̅̅
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
52
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
53