## 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

- Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.

- Prove that ∠VYX ≅ ∠WYZ in the given figure.

- Prove that¯FL≅¯HN in the given diagram.

- Prove that¯FL≅¯HN in the given diagram.

- Prove that ¯AC≅¯GE in the given diagram.

- Write a two-column proof from the given diagram.

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

**Given: **¯MN≅¯KN,∠PMN≅∠NKL

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

**Given: **TS≅¯TV,¯SR≅¯VW

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

m∠A=x°;m∠B=(4x)° and m∠C=(5x)°.

- 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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