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
- 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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