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Unit 5 – Triangle Congruence

Day Classwork Homework

Wednesday

10/25 Unit 4 Test D1 - Proving SAS through Rigid Motions

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Thursday

10/26

Proving SAS through Rigid Motions D2 - Using SAS to Prove Triangles

Congruent CPCTC

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Friday

10/27

Using SAS to Prove Triangles

Congruent CPCTC

D3 - Proving ASA and SSS through Rigid

Motions

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Monday

10/30

Proving ASA and SSS through Rigid

Motions

D4 - AAS, HL

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Tuesday

10/31

AAS, HL

Unit 5 Quiz 1

D5 - Adding and Subtracting

Segments/Angles

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Wednesday

11/1

Adding and Subtracting

Segments/Angles

D6 - Overlapping Triangle proofs

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Thursday

11/2

Overlapping Triangle proofs

Unit 5 Quiz 2

D7 - Double Triangle Proofs

Watch Video

Friday

11/3

Double Triangle Proofs Review

Monday

11/6

Review

Unit 5 Quiz 3

Review

Tuesday

11/7

Review

Study

Wednesday

11/8 Unit 5 Test

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B'''

A

B

C

Day 1 – Proving SAS through Rigid Motions

Recall: We have agreed to use the word ‘congruent’ to mean ‘there exists a _________________________ of basic rigid motions of the plane that maps one figure to the other.’

We are going to show that there are criteria that refer to a few parts of the two triangles and a correspondence between them that guarantee congruency (i.e., existence of rigid motion). We start with the Side-Angle-Side (SAS) criteria.

In the figure below, 2 pairs of sides are congruent and the included angles are congruent.

AB ____________ AC __________ A ___________

There are 3 rigid motions that will map ABC to ' ' 'A B C . Describe each rigid motion in words and

symbols .

1.

2.

3.

3

B"

C"

A

B

C

B'''

A

B

C

Examples:

1.

Case Diagram Transformations Needed

Shared Side

Shared Vertex

2. Given: Triangles with a pair of corresponding sides of equal length and a pair of included angles of equal measure. Sketch and label three phases of the sequence of rigid motions that prove the two triangles to be congruent.

Transformation

Sketch

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Directions: Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of sides or angles are congruent and why. If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle onto the other.

1. Given: ∠𝐿𝑁𝑀 ≅ ∠𝐿𝑁𝑂, 𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑂𝑁̅̅ ̅̅ .

Do △ 𝐿𝑀𝑁 and △ 𝐿𝑂𝑁 meet the SAS criteria?

2. Given: ∠𝐻𝐺𝐼 ≅ ∠𝐽𝐼𝐺, 𝐻𝐺̅̅ ̅̅ ̅ ≅ 𝐽�̅�.

Do △ 𝐻𝐺𝐼 and △ 𝐽𝐼𝐺 meet the SAS criteria?

3. Given: 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ .

Do △ 𝐴𝐵𝐷 and △ 𝐶𝐷𝐵 meet the SAS criteria?

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4. Given: 𝑚∠𝑅 = 25°, 𝑅𝑇 = 7", 𝑆𝑈 = 5", 𝑆𝑇 = 5".

Do △ 𝑅𝑆𝑈 and △ 𝑅𝑆𝑇 meet the SAS criteria?

5. Given: 𝐾𝑀̅̅ ̅̅ ̅ and 𝐽𝑁̅̅̅̅ bisect each other.

Do △ 𝐽𝐾𝐿 and △ 𝑁𝑀𝐿 meet the SAS criteria?

6. Given: 𝐴𝐸̅̅ ̅̅ bisects angle ∠𝐵𝐶𝐷, 𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ .

Do △ 𝐶𝐴𝐵 and △ 𝐶𝐴𝐷 meet the SAS criteria?

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Day 2 – Using SAS to Prove Triangles Congruent

Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria to prove the triangles are congruent.

1. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅

__________________________

__________________________

Prove: △ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵.

2. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅

AB RS

__________________________

__________________________

Prove: △ 𝐴𝐵𝐶 ≅△ 𝑅𝑆𝑇.

Example:

We already know that the base angles of an isosceles triangles are congruent. We are going to prove this fact in two ways: (1) by using transformations, and (2) by using SAS triangle congruence criteria.

Prove Base Angles of an Isosceles are Congruent Using Transformations

Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ .

Prove: ∠𝐵 ≅ ∠𝐶.

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Prove Base Angles of an Isosceles are Congruent Using SAS

Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ .

Prove: ∠𝐵 ≅ ∠𝐶.

CPCTC Corresponding parts of congruent

triangles are congruent.

1. Given: EA AD

FD AD

AC DB

AE DF

Prove: EC FB

A

E

D B C

F

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2. Given: KM bisects LKJ

LK JK

Prove: M is the midpoint of LJ

3. Given: KM LJ

LM JM

Prove: KLJ is isosceles

L

K

J M

L

K

J M

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4. Given: C is a midpoint of AE

BC DC

1 2

BF DF

Prove: AF EF

5. Given: TR TS

TM TN

Prove: 1 2

A 1

D C B E

F

3 2 4

T

R

1

S

2

M N

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Day 3 – Proving ASA and SSS through Rigid Motions

A second criteria that guarantees congruency (i.e., existence of rigid motion) is Angle-Side-Angle.

1. In the diagram below, describe the rigid motion that maps ABC to '''ABC . Will the triangles be

congruent by SAS? Why or why not?

Since the triangles have two pairs of corresponding congruent angles and the included side congruent, the triangles are congruent by Angle-Side-Angle.

2. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the ASA triangle congruence criteria to prove the triangles are congruent.

Given: ABC DCB

__________________________

__________________________

Prove: △ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵.

A

Given: AD BC

__________________________

__________________________

Prove: △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷

B D C

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3. In the diagram below, describe the rigid motion that maps ABC to 'AB C . Will the triangles be

congruent? Why or why not?

Since the triangles have three pairs of corresponding congruent sides, the triangles are congruent by Side-Side-Side.

4. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SSS triangle congruence criteria to prove the triangles are congruent.

Given: KM and JN bisect each other at L

__________________________

Prove: △ 𝐽𝐾𝐿 ≅△ 𝑁𝑀𝐿.

Given: R is the midpoint of KL

__________________________

__________________________

Prove: △ 𝐽𝐾𝑅 ≅△ 𝐽𝐿𝑅

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1. Given: 𝑀 is the midpoint of 𝐻𝑃̅̅ ̅̅ , ∠𝐻 ≅ ∠𝑃.

Prove: GHM RPM

2. Given: Rectangle 𝐽𝐾𝐿𝑀 with diagonal 𝐾𝑀̅̅ ̅̅ ̅.

Prove: JKM LMK

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3. Given: Circles with centers 𝐴 and 𝐵 intersect at 𝐶 and 𝐷.

Prove: ∠𝐶𝐴𝐵 ≅ ∠𝐷𝐴𝐵.

4. Given: ∠𝑤 ≅ ∠𝑥 and ∠𝑦 ≅ ∠𝑧.

Prove: (1) △ 𝐴𝐵𝐸 ≅△ 𝐴𝐶𝐸. (2) 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅ .

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Day 4 –AAS & HL Triangle Congruence

1. Consider a pair of triangles that meet the AAS criteria (see below). If you knew that two angles of one triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each triangle?

Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are congruent?

Therefore, the AAS criterion is actually an extension of the triangle congruence criterion.

Practice: Using only the information given, decide whether the triangles are congruent by ASA or AAS.

1. AB BC

DC BC

2.

BC AD

AB DC

3. 4.

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5. Consider the two right triangles below that have congruent hypotenuses and one pair of congruent legs. Are the triangles congruent? Why?

This is our last criteria to prove triangles congruent: Hypotenuse-Leg (HL)

**Note that only RIGHT triangles can be proved congruent using HL**

Practice: Using only the information given, decide by which method the triangles could be proved congruent.

1. 2. ED BA

AC DF

3. HG HI 4. E is the midpoint of CD .

A and B are right angles

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Criteria that do not determine two triangles as congruent:

Two sides and a non-included angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚. Yet, the triangles are not congruent.

Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations.

The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be congruent, but they might not be; therefore we cannot categorize SSA as congruence criterion.

Practice: Decide if the triangles are congruent. If so, by what method?

1. Given: A X , BA YX , ZX CA

2. Given: CA ZX , BC YZ , A X

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Three Congruent Angles (AAA): Observe the diagrams below. What is the measure of the missing angles? Notice that the triangles have 3 pairs of congruent angles. Yet, the triangles are not congruent.

The pattern of AAA cannot guarantee congruence criteria. In other words, two triangles under AAA criteria might be congruent, but they might not be; therefore we cannot categorize AAA as congruence criterion.

1. Given: BD AC A C Prove: ABD CBD

2. Given: PQ RS

QO RO

Prove: XO YO

A

B

C D

R

Q

2 S Y

O

P X

1

45°

45°

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Day 5 - ADDING & SUBTRACTING SEGMENTS & ANGLES

If a pair of congruent segments/angles are added to another pair of congruent segments/angles, then the

resulting segments/angles are congruent. Similarly, if a pair of congruent segments/angles are subtracted

from a pair of congruent segments/angles, then the resulting segments/angles are congruent.

Statements Reasons___________

1. AB CD 1. Given

2. BC BC 2. Reflexive Property

3. AB BC CD BC 3. Addition Property

4. AC AB BC 4. A whole =’s the sum of its BD CD BC parts 5. AC BD 5. Substitution

Statements Reasons___________

1. AC BD 1. Given

2. BC BC 2. Reflexive Property

3. AC AB BC 3. A whole =’s the sum of its BD CD BC parts

4. AB BC CD BC 4. Substitution 5. AC BD 5. Subtraction Property

Examples

1.

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2.

3.

4.

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5.

6.

7.

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8. Given: PS UR

PQ UT

QR TS

Prove: Q T

9. Given: 1 4

2 3

Prove: 5 6

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Day 6 & 7 – Overlapping and Double Triangle Proofs

1. Given: 𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ ⊥ 𝐷𝐶̅̅ ̅̅

𝐷𝐵̅̅ ̅̅ bisects ∠𝐴𝐵𝐶, 𝐴𝐶̅̅ ̅̅ bisects ∠𝐷𝐶𝐵

𝐸𝐵̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅

Prove: △ 𝐵𝐸𝐴 ≅ ∆𝐶𝐸𝐷

2. Given: 𝐵𝐹̅̅ ̅̅ ⊥ 𝐴𝐶̅̅ ̅̅ , 𝐶𝐸̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅

𝐴𝐸̅̅ ̅̅ ≅ 𝐴𝐹̅̅ ̅̅

Prove: △ 𝐴𝐶𝐸 ≅ ∆𝐴𝐵𝐹

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3. Given: 𝑋𝐽̅̅ ̅ ≅ 𝑌𝐾̅̅ ̅̅ , 𝑃𝑋̅̅ ̅̅ ≅ 𝑃𝑌̅̅ ̅̅ , ∠𝑍𝑋𝐽 ≅ ∠𝑍𝑌𝐾

Prove: 𝐽𝑌̅̅ ̅ ≅ 𝐾𝑋̅̅ ̅̅

4. Given: 𝐽𝐾̅̅ ̅ ≅ 𝐽�̅�, 𝐽𝐾̅̅ ̅ ∥ 𝑋𝑌̅̅ ̅̅

Prove: 𝑋𝑌̅̅ ̅̅ ≅ 𝑋𝐿̅̅̅̅

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5. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4

Prove: 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅

6. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅

Prove: (a) △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷

(b) ∠5 ≅ ∠6

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7. Given: 𝑨𝑩̅̅ ̅̅ ≅ 𝑨𝑪̅̅ ̅̅ ,

𝑅𝐵̅̅ ̅̅ ≅ 𝑅𝐶̅̅ ̅̅ ,

Prove: 𝑆𝐵̅̅̅̅ ≅ 𝑆𝐶̅̅̅̅

8. Given: 𝐽𝐾̅̅ ̅ ≅ 𝐽�̅�, 𝐽𝑋̅̅ ̅ ≅ 𝐽𝑌̅̅ ̅

Prove: 𝐾𝑋̅̅ ̅̅ ≅ 𝐿𝑌̅̅̅̅

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9. Given: 𝐴𝐷̅̅ ̅̅ ⊥ 𝐷𝑅̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝑅̅̅ ̅̅ ,

𝐴𝐷̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅

Prove: ∠𝐷𝐶𝑅 ≅ ∠𝐵𝐶𝑅