Corollaries of the Angle-Sum Property of a Triangle Revision Notes for Leaving Cert Mathematics

Corollaries of the Angle-Sum Property of a Triangle

Overview

The Angle-Sum Property of a Triangle states that the sum of the three interior angles of any triangle is always $180^\circ$ . This fundamental property leads to several important corollaries that describe relationships between angles and sides in specific types of triangles.

Corollary 1: Isosceles Triangle

Statement: If two angles in a triangle are equal, the triangle is isosceles.
Why It Works:
- If two angles are equal, the sides opposite those angles must also be equal, making the triangle isosceles.
- This follows directly from the Angle-Sum Property since the third angle is uniquely determined by the other two angles.

Corollary 2: Acute Angles in a Right Triangle

Statement: In a right triangle, the two acute angles are complementary, meaning their measures add up to $90^\circ$
Why It Works:
- A right triangle has one angle equal to $90^\circ$
- By the Angle-Sum Property:

\text{Sum of angles} = 180^\circ = 90^\circ (\text{right angle}) + \text{sum of acute angles}

Therefore, the two acute angles must add up to $90^\circ$

Worked Examples

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Example 1: Using Corollary 1

Problem: A triangle has two angles measuring $50^\circ$ and $50^\circ$

Prove that the triangle is isosceles.

Solution:

Step 1: The two angles are equal, so by Corollary 1, the sides opposite these angles are also equal.

Step 2: The triangle is therefore isosceles.

Answer: The triangle is isosceles.

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Example 2: Using Corollary 2

Problem: In a right triangle, one of the acute angles measures $35^\circ$

Find the other acute angle.

Solution:

Step 1: The two acute angles are complementary:

\text{Other angle} = 90^\circ - 35^\circ = 55^\circ

Answer: The other acute angle is $55^\circ$

Summary

Angle-Sum Property: The sum of the angles in a triangle is $180^\circ$
Corollary 1: If two angles are equal, the triangle is isosceles, as the sides opposite those angles are also equal.
Corollary 2: In a right triangle, the two acute angles are complementary (add up to $90^\circ$ ). These corollaries simplify the analysis and classification of triangles in geometry.

Corollaries of the Angle-Sum Property of a Triangle (Leaving Cert Mathematics): Revision Notes