## Key Concepts

- Use the SAS congruence postulate

## Introduction

### Side-Angle-Side (SAS) congruence postulate:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

### Prove Triangles Congruent by SAS and HL

**Example 1:** Use the SAS congruence postulate to prove that

**Solution:**

**Given:**

**To Prove:**

**Proof:**

**Example 2:** Prove that form the given figure.

**Solution:**

**Given:**

**To Prove:**

### What is a right angle?

When two straight lines are perpendicular to each other at point of intersection, they form a right angle. It is denoted by the symbol ∟

.

Examples of right angles include different shapes of a polygon.

#### Real-life examples

Real-world examples of a right angle are, the corners of a room, window, etc.

### What is a right triangle?

When one of the interior angles of a triangle is 90° it is called a right triangle.

From the above figure, the longest side of the triangle is the hypotenuse and the two opposite sides are the height and the base.

#### Types of right triangles:

**Acute triangles**– all angles measures less than 90°.**Obtuse triangles****–**one angle measures between 90° and 180°.**Equilateral triangles****–**all angles measure 60°.**Right triangles**– one angle measures exactly 90°.

### What is Hypotenuse Leg (HL)?

In a right triangle, the two sides adjacent to the right angle are called legs and the side opposite to the right angle is called the hypotenuse.

#### Hypotenuse Leg Congruent Theorem (HL):

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

**Given:**

**To Prove:**

**Example 3: **If PR ⊥ QS*, *prove that ∆PQR* *and ∆PRS are congruent.

**Solution:**

Given, ΔPQR and ΔPRS are right triangles, since 90° angle at point R.

**Proof:**

**Example 4:** Using the Hypotenuse Leg (HL) congruence theorem, prov that

## Exercise

- Prove that from the given figure

- Prove that in the given figure.

- Prove that from the given figure.

- Prove that from the given figure.

- Prove that from the given figure.

- Prove that from the given figure.

- Prove that from the given figure.

- If PQRS is a square and ∆SRT is an equilateral triangle, then prove that PT = QT.

- Prove that the medians of an equilateral triangle are equal.

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

### What have we learned

- Understand and apply the SAS congruence postulate.
- Identify the types of right triangles.
- Identify the properties of right triangles.
- Understand and apply the HL congruence theorem.
- Solve the problems on SAS congruence of triangles.
- Solve the problems on HL congruence of triangles.

### Summary

**Types of right triangles:**

**Acute triangles**– all angles measures less than 90°.**Obtuse triangles –**one angle measures between 90° and 180°**Equilateral triangles –**all angles measure 60°.**Right triangles**– one angle measures exactly 90°.

**Side-Angle-Side (SAS) congruence postulate:**

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

**Hypotenuse Leg Congruent Theorem (HL):**

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

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