Questions: The m∠B is 86°, the measure of arc BC is 64°, and the measure of arc CD is 109°. Find the measure of <C.

The m∠B is 86°, the measure of arc BC is 64°, and the measure of arc CD is 109°. Find the measure of <C.
Transcript text: Geometry - 412200CW UNIT 13 LESSON 11 Circles Circles Review Back to Intro Page Question 6. The $\mathrm{m}_{\angle} \mathbf{B}$ is $86^{\circ}$, the measure of arc $\mathbf{B C}$ is $64^{\circ}$, and the measure of arc $C D$ is $109^{\circ}$. Find the measure of $<\mathrm{C}$.
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Solution

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Solution Steps

Step 1: Find the measure of arc AD.

The sum of the measures of the arcs of a circle is 360°. We are given that arc BC = 64° and arc CD = 109°. We are also given that angle B is 86°. Since quadrilateral ABCD is inscribed in the circle, the sum of opposite angles is 180°. Thus, angle D = 180° - 86° = 94°.

Since the measure of an inscribed angle is half the measure of its intercepted arc, we have that 2 * angle D = arc ABC. Therefore, arc ABC = 2 * 94° = 188°. We also know that arc ABC = arc AB + arc BC. Thus, arc AB = arc ABC - arc BC = 188° - 64° = 124°.

Similarly, 2 * angle B = arc ADC. Thus, arc ADC = 2 * 86° = 172°. Also, arc ADC = arc AD + arc CD, so arc AD = arc ADC - arc CD = 172° - 109° = 63°.

Alternatively, since arc BC + arc CD + arc AB + arc AD = 360°, arc AD = 360° - 64° - 109° - 124° = 63°.

Step 2: Find the measure of angle C.

The measure of inscribed angle C is equal to half the measure of its intercepted arc, which is arc DAB, or arc AD + arc AB. Therefore, angle C = (arc AD + arc AB)/2 = (63° + 124°)/2 = 187°/2 = 93.5°.

Final Answer:

The measure of angle C is 93.5°.

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