## 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

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: