SAS Congruence

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

1. Prove that from the given figure

2. Prove that in the given figure.

3. Prove that from the given figure.

4. Prove that  from the given figure.

5. Prove that  from the given figure.

6. Prove that from the given figure.

7. Prove that  from the given figure.

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

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

10. 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.