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.

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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.

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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 $90^\circ$

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Worked Examples

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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.