Chapter 6.CR, Problem 30CR

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

To determine

**To Find: **The measures of

Explanation

**Concept:**

**Definition:**

A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are radii of the circle.

**Definition:**

An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle.

**Theorem 6.1.2:**

The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

**Theorem 6.1.8:**

Congruent chords are located at the same distance from the center of a circle.

**Theorem 6.1.9:**

Angle inscribed in a semicircle is a right angle.

**Theorem 6.2.2:**

The measures of an angle formed by two chords that intersect within a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

**Theorem 6.2.4:**

The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc.

**Theorem 6.2.5:**

The measure of an angle formed when two secants at a point outside the circle is one-half the difference of the measures of the two intercepted arcs.

**Calculation:**

Given

Since

And

By **Theorem 6.1.9:**

Angle inscribed in a semicircle is a right angle.

So,

By **Theorem 6.2.4:**

The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc.

So,

Also,

Since sum of the three angles in

And

And

By **Theorem 6.2.2:**

The measures of an angle formed by two chords that intersect within a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle

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