Vertex Angle Study Guide

Definition:

The vertex angle is the angle formed by the two lines or rays that extend from the same vertex. In the context of geometry, it is often used to refer to the angle formed in a polygon at its vertex.

Properties of Vertex Angles:

• Vertex angles in a triangle: In a triangle, the sum of all three vertex angles is always 180 degrees.
• Vertex angles in a polygon: In a regular polygon (a polygon with all sides and angles equal), the measure of each vertex angle is given by the formula: (n-2) * 180 / n, where n is the number of sides in the polygon.

Examples:

1. In a triangle ABC, if angle A = 60 degrees and angle B = 40 degrees, then the measure of angle C = 180 - 60 - 40 = 80 degrees.

2. In a regular hexagon (a polygon with 6 sides), the measure of each vertex angle = (6-2) * 180 / 6 = 120 degrees.

Practice Problems:

1. In a quadrilateral ABCD, if angle A = 70 degrees, angle B = 110 degrees, and angle C = 90 degrees, what is the measure of angle D?
2. Find the measure of each vertex angle in a regular pentagon (5 sides).

Solution to Practice Problems:

1. Using the property that the sum of all angles in a quadrilateral is 360 degrees, angle D = 360 - 70 - 110 - 90 = 90 degrees.
2. Each vertex angle in a regular pentagon = (5-2) * 180 / 5 = 108 degrees.

Summary:

The vertex angle is an important concept in geometry, particularly in the study of polygons and triangles. Understanding the properties and measurements of vertex angles is essential for solving various geometric problems.