What is m CD ? 78° B A M 106° D O A. 92°

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
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Question 36

### Finding the Measure of Arc \( \overset{\frown}{CD} \)

Given a circle with points \( A \), \( B \), \( C \), and \( D \) on its circumference and the following angles:
- \( \angle AMB = 78^\circ \)
- \( \angle CMD = 106^\circ \)

We are asked to find the measure of arc \( \overset{\frown}{CD} \).

#### Diagram Explanation:
- The circle is marked with points \( A \), \( B \), \( C \), and \( D \).
- \( M \) is the intersection of the lines \( \overline{AB} \) and \( \overline{CD} \) inside the circle.
- The angle formed at \( M \) by arc \( \overset{\frown}{AB} \) is \( 78^\circ \).
- The angle formed at \( M \) by arc \( \overset{\frown}{CD} \) is \( 106^\circ \).

#### Multiple Choice Options:
- A. \( 92^\circ \)
- B. \( 106^\circ \)
- C. \( 134^\circ \)
- D. \( 1184^\circ \)

To find the correct measure of \( \overset{\frown}{CD} \), we observe the given angles and apply them appropriately to find the relationship between the angles and arcs in the circle.

---
### Explanation Steps:
Using theorem properties of inscribed angles and their intercepted arcs, we can derive the required information to answer the question. Note that the sum of the measures of angles formed by intersecting chords is half the sum of the measures of the arcs they intercept.

Now, let's proceed with solving the problem using appropriate circle theorems to arrive at the measure for arc \( \overset{\frown}{CD} \).

In this case, since \( \angle CMD \) intercepts \( \overset{\frown}{CD} \):

  \[
 m \angle CMD = \frac{1}{2}( \overset{\frown}{CD} + \overset{\frown}{AB})
  \]
  
Given \( m\angle CMD = 106^\circ \) and assuming \( \overset{\frown}{AB} = 78^\circ \) 
So we put these values in formula to find \( \overs
Transcribed Image Text:### Finding the Measure of Arc \( \overset{\frown}{CD} \) Given a circle with points \( A \), \( B \), \( C \), and \( D \) on its circumference and the following angles: - \( \angle AMB = 78^\circ \) - \( \angle CMD = 106^\circ \) We are asked to find the measure of arc \( \overset{\frown}{CD} \). #### Diagram Explanation: - The circle is marked with points \( A \), \( B \), \( C \), and \( D \). - \( M \) is the intersection of the lines \( \overline{AB} \) and \( \overline{CD} \) inside the circle. - The angle formed at \( M \) by arc \( \overset{\frown}{AB} \) is \( 78^\circ \). - The angle formed at \( M \) by arc \( \overset{\frown}{CD} \) is \( 106^\circ \). #### Multiple Choice Options: - A. \( 92^\circ \) - B. \( 106^\circ \) - C. \( 134^\circ \) - D. \( 1184^\circ \) To find the correct measure of \( \overset{\frown}{CD} \), we observe the given angles and apply them appropriately to find the relationship between the angles and arcs in the circle. --- ### Explanation Steps: Using theorem properties of inscribed angles and their intercepted arcs, we can derive the required information to answer the question. Note that the sum of the measures of angles formed by intersecting chords is half the sum of the measures of the arcs they intercept. Now, let's proceed with solving the problem using appropriate circle theorems to arrive at the measure for arc \( \overset{\frown}{CD} \). In this case, since \( \angle CMD \) intercepts \( \overset{\frown}{CD} \): \[ m \angle CMD = \frac{1}{2}( \overset{\frown}{CD} + \overset{\frown}{AB}) \] Given \( m\angle CMD = 106^\circ \) and assuming \( \overset{\frown}{AB} = 78^\circ \) So we put these values in formula to find \( \overs
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