Geometry

Question Analysis

This question involves the properties of inscribed angles in a circle. The key is to first use the property that an inscribed angle in a semicircle is a right angle to find the measure of , and then use the relationship between inscribed angles and the arcs they intercept to find the measure of arc .

 

Key Concept Explanation

1. Inscribed angle in a semicircle: An inscribed angle subtended by a diameter (i.e., an inscribed angle in a semicircle) is a right angle, equal to .

2. Relationship between inscribed angle and intercepted arc: The measure of an inscribed angle is half of the measure of the arc it intercepts.

 

Step - by - Step Solution

1. Find the measure of

Since  is the diameter of the circle, by the property that the inscribed angle in a semicircle is a right angle, .

Given , in , using the fact that the sum of interior angles of a triangle is , we can calculate .

2. Find the measure of arc

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