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Congruence of Triangles

Site: Saylor Academy
Course: GKT101: General Knowledge for Teachers – Math
Book: Congruence of Triangles
Printed by: Guest user
Date: Saturday, May 18, 2024, 1:06 AM

Description

Proving if some shapes, especially triangles, are congruent is an important part of the study of geometry. This lecture series discusses the three theorems that establish the congruence of triangles.

Table of contents

Determining congruent triangles

Calculating angle measures to verify congruence

Corresponding parts of congruent triangles are congruent

Proving triangle congruence

Practice

   

Determine congruent triangles - Questions

1. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

2. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

3. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

4. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

5. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

6. Consider the two triangles shown below.



Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.

7. Consider the two triangles shown below.


Note: The triangles are not drawn to scale.

Are the two triangles congruent?

Choose 1 answer:

(A) Yes

(B) No

(C) There is not enough information to say.


Determine congruent triangles - Answers

1. No, the triangles are not congruent.

2. There is not enough information to say whether or not the triangles are congruent.

3. Yes, the triangles are congruent.

4. No, the triangles are not congruent.

5. No, the triangles are not congruent.

6. There is not enough information to say whether or not the triangles are congruent.

7. Yes, the triangles are congruent.

Prove triangle congruence - Questions

1. Holden tried to prove that \triangle F G H \cong \triangle F I H.

 

  Statement Reason
1 \overline{I H} \cong \overline{G H} Given
2 \overline{F H} \cong \overline{F H} Line segments are congruent to themselves. 
3 \triangle F G H \cong \triangle F I H Side-side congruence

 

What is the first error Holden made in his proof?

Choose 1 answer:

(A) Holden used an invalid reason to justify the congruence of a pair of sides or angles.

(B) Holden only established some of the necessary conditions for a congruence criterion.

(C) Holden established all necessary conditions, but then used an inappropriate congruence criterion.

(D) Holden used a criterion that does not guarantee congruence.

2. Prove that \triangle P Q R \cong \triangle S T R.

 

 

Statement

Reason

1

\overline{Q R} \cong \overline{T R}

Given

2

\overline{P R} \cong \overline{S R}

Given

3

Pick statement (\angle P Q R \cong \angle S T R / \angle R P Q \cong \angle R S T / \angle Q R P \cong \angle T R S)

Pick congruence criterion (Angle-angle-side/Angle-side-angle/Side-angle-side/Side-side-side)

4

\triangle P Q R \cong \triangle S T R

Pick reason (Given/Vertical angles are congruent./Corresponding parts of congruent triangles are congruent).

 

3. Prove that \triangle A B C \cong \triangle A D C.

 

Statement

Reason

1

m \angle B A C=m \angle D A C=51^{\circ}

Pick statement (Given / Vertical angles are congruent. / Linear pair angles are supplementary).

2

A B=A D=4

Given

3

Pick statement ( A B = A B / A C = A C / C D = C D )

Segments are the same length as themselves.

4

\triangle A B C \cong \triangle A D C

Pick congruence criterion (angle-angle-side / Angle-side-angle / Side-angle-side / Side-side-side)

 

4. Jordy tried to prove that \triangle A B E \cong \triangle B C D.

 

 

Statement

Reason

1

\angle B C D \cong \angle A B E

Given

2

\angle C D B \cong \angle B E A

Given

3

\overleftarrow{B D} \| \overleftrightarrow{A E}

Given

4

\angle C B D \cong \angle B A E

Corresponding angles on parallel lines are congruent.

5

\triangle A B E \cong \triangle B C D

Angle-angle-angle congruence

 

What is the first error Jordy made in his proof?

Choose 1 answer:

(A) Jordy used an invalid reason to justify the congruence of a pair of sides or angles.

(B) Jordy only established some of the necessary conditions for a congruence criterion.

(C) Jordy established all necessary conditions, but then used an inappropriate congruence criterion.

(D) Jordy used a criterion that does not guarantee congruence.

Prove triangle congruence - Answers

1. Holden used a criterion that does not guarantee congruence.

2.

 

Statement

Reason

1

\overline{Q R} \cong \overline{T R}

Given

2

\overline{P R} \cong \overline{S R}

Given

3

\angle Q R P \cong \angle T R S

Vertical angles are congruent.

4

\triangle P Q R \cong \triangle S T R

Side-angle-side congruence

 

3.

  Statement Reason
1 m \angle B A C=m \angle D A C=51^{\circ} Given.
2 A B=A D=4 Given
3 A C=A C Segments are the same length as themselves.
4 \triangle A B C \cong \triangle A D C Side-angle-side congruence


4. Jordy used a criterion that does not guarantee congruence.