   Chapter 6.CR, Problem 30CR Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

G i v e n :                       ⊙ O   w i t h   d i a m e t e r   A C -   a n d   t a n g e n t   D E ⃡                                                                       m A D ^ = 136 °   a n d   m B C ^ = 50 °                 F i n d :                           T h e   m e a s u r e s   o f   ∠ 1   t h r o u g h   ∠ 10 To determine

To Find: The measures of 1 through 10, if O with diameter AC- and tangent DE. mAD^=136° and mBC^=50°. 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 O with diameter AC- and tangent DE. mAD^=136° and mBC^=50°.

Since mBC^=50°, its central angle is 9=50°.

By Theorem 6.1.9:

Angle inscribed in a semicircle is a right angle.

So, 5=90°

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, 6=12mDC^ and 4=12AOD=12×136°=68°

Also, 7=12AOD=12×136°=68°

Since sum of the three angles in ADC=180°

5+7+8=180°.

90+68+8=180°.

158°+8=180°.

8=180°-158°

8=22°

And 6=12mDC^=442=22°

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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