Question
The m∠B is 86°, the measure of arc BC is 64°, and the measure of arc CD is 109°. Find the measure of <D. (1 point) Responses 64° 64° 86.5° 86.5° 93.5° 93.5° 94°
Answers
Answered by
GPT-4o mini
To find the measure of angle \( D \) in this scenario, we can use the properties of inscribed angles and arcs.
Given:
- \( m\angle B = 86^\circ \)
- \( m\) arc \( BC = 64^\circ \)
- \( m\) arc \( CD = 109^\circ \)
To find \( m\angle D \), we need to determine which arcs define angle \( D \)'s measure.
### Step 1: Find \( m\angle D \)
In a circle, the measure of an inscribed angle is equal to half the measure of the intercepted arc. Therefore, we first need to determine which arcs are relevant for \( \angle D \).
1. **Intercepted Arc for \( \angle D \)**:
- If angle \( D \) intercepts arc \( BC \) and arc \( CD \), the angle formed would then intercept arc \( BD \).
- Thus, the measure of arc \( BD \) can be found by adding the measures of arcs \( BC \) and \( CD \):
\[
m\text{ arc } BD = m\text{ arc } BC + m\text{ arc } CD = 64^\circ + 109^\circ = 173^\circ
\]
2. **Calculate \( m\angle D \)**:
- Now, since \( \angle D \) intercepts the arc \( BD \), we find \( \angle D \) using the formula:
\[
m\angle D = \frac{1}{2} m\text{ arc } BD
\]
Substituting the measure we found:
\[
m\angle D = \frac{1}{2} \times 173^\circ = 86.5^\circ
\]
Thus, the measure of angle \( D \) is \( 86.5^\circ \).
### Conclusion:
Out of the options provided, the correct answer is \( 86.5^\circ \).
Given:
- \( m\angle B = 86^\circ \)
- \( m\) arc \( BC = 64^\circ \)
- \( m\) arc \( CD = 109^\circ \)
To find \( m\angle D \), we need to determine which arcs define angle \( D \)'s measure.
### Step 1: Find \( m\angle D \)
In a circle, the measure of an inscribed angle is equal to half the measure of the intercepted arc. Therefore, we first need to determine which arcs are relevant for \( \angle D \).
1. **Intercepted Arc for \( \angle D \)**:
- If angle \( D \) intercepts arc \( BC \) and arc \( CD \), the angle formed would then intercept arc \( BD \).
- Thus, the measure of arc \( BD \) can be found by adding the measures of arcs \( BC \) and \( CD \):
\[
m\text{ arc } BD = m\text{ arc } BC + m\text{ arc } CD = 64^\circ + 109^\circ = 173^\circ
\]
2. **Calculate \( m\angle D \)**:
- Now, since \( \angle D \) intercepts the arc \( BD \), we find \( \angle D \) using the formula:
\[
m\angle D = \frac{1}{2} m\text{ arc } BD
\]
Substituting the measure we found:
\[
m\angle D = \frac{1}{2} \times 173^\circ = 86.5^\circ
\]
Thus, the measure of angle \( D \) is \( 86.5^\circ \).
### Conclusion:
Out of the options provided, the correct answer is \( 86.5^\circ \).