GEOMETRYHELP

Notes Lesson 5.2

Congruent Triangles

Target 4.1

GEOMETRYHELP

Definition:Congruent triangles are triangles that have all corresponding sides congruent and all corresponding angles congruent.

Congruent FiguresLesson 5.2

Write a congruence statement for the triangles at the left.

What information is sufficient to prove triangles congruent?

GEOMETRYHELP

XYZ KLM,

YZ= x + 10

LM= 2x

Find the value

of x and the

lengths of the

given sides.

Begin marking these triangles with corresponding angles that are congruent.

Congruent Figures

Target 4.1Example 2:

GEOMETRYHELP

Leading to Target 4.2

GEOMETRYHELP

GEOMETRYHELP

GEOMETRYHELP

Triangle Congruence by SSS and SASLesson 5.3

GEOMETRYHELP

Write a two-column proof.

Given: A B, AP BP

Prove: APX BPY

Triangle Congruence by ASA and AAS

Statements Reasons

1. 1. Given BPAPBA ,

2. 2. Vertical angles are congruent.

3. 3. ASABPYAPX

1 21 2

Example 3:

GEOMETRYHELP

Write a two-column proof that uses AAS. Given: B D, AB || CDProve: ABC CDA

Statements Reasons

1. B D, AB || CD 1. Given

4. ABC CDA 4. AAS Theorem

2. 1 2 2. If lines are ||, then alternate interior angles are .

3. AC CA 3. Reflexive Property of Congruence

Triangle Congruence by ASA and AAS

1

2

Given

Not an included side

TARGET 4.3 & 4.5

Example 4:

GEOMETRYHELP

From the given information, can you prove that the triangles are congruent. Explain.

Given: M is the midpoint of XY, AX AY

Prove: AMX AMY

Copy the diagram. Mark the congruent sides.

Midpoint M implies MX MY.

AM AM by the Reflexive

Property of Congruence.

AMX AMY by the

SSS Postulate.

Triangle Congruence by SSS and SAS

Target 4.2Example 5:

GEOMETRYHELP

b) AD BC. What other information do you need to prove ADC BCD by SAS?

DC CD by the Reflexive Property.You now have two pairs of corresponding congruent sides. Therefore if you know ADC BCD, you can prove ADC BCD by SAS.

Triangle Congruence by SSS and SAS

a) Draw the two congruent triangles separately.

Target 4.2A

D C D

B

C

a)

b)

Example 6:

GEOMETRYHELP

YOU TRY #1TARGET 4.3

GEOMETRYHELP

Review: Right triangle:

Leg

Leg

Hypotenuse

L

L

H

H

L

L

TARGET 4.4

GEOMETRYHELP

What additional information is needed to prove.Prove: ABC DCBby HL.

Since BC CB Reflexive Property of Congruence

To prove ABC DCB by the (HL Theorem).

Congruence in Right Triangles

TARGET 4.4

C D

B B

C

A

You must prove that BD CA

Example 7:

GEOMETRYHELP

One student wrote “ CPA MPA

by SAS” for the diagram below.

Is the student correct? Explain.

The congruent angles are not included between the corresponding congruent sides.The triangles are not congruent by the SAS Postulate, but they are congruent by the HL Theorem.

The diagram shows the following congruent parts.

CA MA

CPA MPA

PA PA

Congruence in Right Triangles

TARGET 4.4Example 8:

GEOMETRYHELP

YOU TRY #2 TARGET 4.4

What additional information will allow you to prove the triangles congruent by the HL theorem?

GEOMETRYHELP

YOU TRY #3 TARGET 4.4