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# Applications of Congruent Triangles

## Introduction

### Applications of Congruent Triangles

Example 1: In the given figure, AB = BC and AD = CD. Show that BD bisects AC at right angles.

Solution:

From the given figure, ∆ABD ≅ ∆CBD

Given: AB = BC and AD = CD

To prove: ∠BEA = ∠BEC = 90° and AE = EC.

Proof:

Now, from ∆ABE and ∆CBE,

Example 2: In a Δ ABC, if AB = AC and ∠ B = 70°, find ∠ A.

Solution:

Given: AB = AC and ∠B = 70°

∠ B = ∠ C [Angles opposite to equal sides are equal]

Therefore, ∠ B = ∠ C = 70°

Sum of angles in a triangle = 180°

∠ A + ∠ B + ∠ C = 180°

Example 3: In the given figure, PQ = PS and ∠QPR = ∠SPR. Prove that ∆ PQR ≅ ∆PSR, Use SAS congruence postulate.

Solution:

Example 4: Identify the congruent triangle in the given figure.

Solution:

Example 5:

Write a 2-column proof for the given figure.

Given: BD is an angle bisector of ∠CDA, ∠C ≅ ∠A

To prove: △CBD ≅ ∠ABD

Solution:

How to prove construction:

The following steps explain the construction of congruent triangles:

Step 1:

To copy∠ A, draw a segment with initial point D. Draw an arc with center A. Using the same radius, draw an arc with center D. Label points B, C, and E.

Step 2:

Draw an arc with radius BC and center E. Label the intersection F.

Step 3:

Draw DF

.

Example 6:

Write a proof to verify that the construction for copying an angle is valid.

Solution:

To prove: Plan for Proof:

Show that △CAB ≅ △FDE, so we can conclude that the corresponding parts ∠A and ∠D are congruent.

Plan in action:

## Exercise

1. Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.
1. Prove that ∠VYX ≅ ∠WYZ in the given figure.
1. Prove that¯FL≅¯HN  in the given diagram.
1. Prove that¯FL≅¯HN  in the given diagram.
1. Prove that ¯AC≅¯GE  in the given diagram.
1. Write a two-column proof from the given diagram.
1. Prove that ∠1 ≅ ∠2 from the given diagram with the given information.

Given: ¯MN≅¯KN,∠PMN≅∠NKL

1. Prove that ∠1 ≅ ∠2 from the given diagram with the given information.

Given: TS≅¯TV,¯SR≅¯VW

1. Find the measure of each angle in the given triangle.

m∠A=x°;m∠B=(4x)° and m∠C=(5x)°.

1. Find the measure of each angle in the given triangle.

m∠A=x°;m∠B=(5x)° and m∠C=(x+19)°.

### What have we learned

• Understand and apply SSS congruence postulate.
• Understand and apply SAS congruence postulate.
• Understand and apply AAS congruence postulate.
• Understand and apply construction proof.
• Solve problems on different congruence of triangles.
• Solve problems on different congruence postulates.

### Concept Map

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