Extra Example 1 Find the measure of each arc of OC, where AB is a diameter. A 65% а. AD b. DAB С. BDA

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
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### Extra Example 1

Find the measure of each arc of circle \( C \), where \( \overline{AB} \) is a diameter.

#### Diagram:
- The diagram shows a circle with center \( C \).
- The line segment \( \overline{AB} \) is the diameter of the circle.
- Point \( D \) is on the circumference of the circle and forms a triangle \( \triangle ADB \) with points \( A \) and \( B \).
- The central angle \( \angle ACD \) is given as \( 65^\circ \).

#### Tasks:
Determine the measure of the following arcs:
a. \( \overset{\frown}{AD} \)
b. \( \overset{\frown}{DAB} \)
c. \( \overset{\frown}{BDA} \)

### Explanation of the Diagram:
- The circle \( C \) has a diameter \( \overline{AB} \), which means \( \angle ACB = 180^\circ \) since a diameter subtends a semicircle.
- The central angle \( \angle ACD \) is given as \( 65^\circ \).
- Since \( \overline{AB} \) is a diameter, \( \overline{ACD} \) and \( \overline{BDC} \) are supplementary and add up to the diameter's total of \( 180^\circ \). Hence, arc \( \overset{\frown}{AD} \) is \( 65^\circ \).

To find the remaining arcs:
- Arc \( \overset{\frown}{BD} \) can be calculated by subtracting \( \overset{\frown}{AD} = 65^\circ \) from \( 180^\circ \), so \( \overset{\frown}{BD} \) is \( 115^\circ \).

Given these details, complete the exercises to find the measures of arcs \( AD, DAB, \) and \( BDA \) based on relationships and supplementary angles in the circle.
Transcribed Image Text:### Extra Example 1 Find the measure of each arc of circle \( C \), where \( \overline{AB} \) is a diameter. #### Diagram: - The diagram shows a circle with center \( C \). - The line segment \( \overline{AB} \) is the diameter of the circle. - Point \( D \) is on the circumference of the circle and forms a triangle \( \triangle ADB \) with points \( A \) and \( B \). - The central angle \( \angle ACD \) is given as \( 65^\circ \). #### Tasks: Determine the measure of the following arcs: a. \( \overset{\frown}{AD} \) b. \( \overset{\frown}{DAB} \) c. \( \overset{\frown}{BDA} \) ### Explanation of the Diagram: - The circle \( C \) has a diameter \( \overline{AB} \), which means \( \angle ACB = 180^\circ \) since a diameter subtends a semicircle. - The central angle \( \angle ACD \) is given as \( 65^\circ \). - Since \( \overline{AB} \) is a diameter, \( \overline{ACD} \) and \( \overline{BDC} \) are supplementary and add up to the diameter's total of \( 180^\circ \). Hence, arc \( \overset{\frown}{AD} \) is \( 65^\circ \). To find the remaining arcs: - Arc \( \overset{\frown}{BD} \) can be calculated by subtracting \( \overset{\frown}{AD} = 65^\circ \) from \( 180^\circ \), so \( \overset{\frown}{BD} \) is \( 115^\circ \). Given these details, complete the exercises to find the measures of arcs \( AD, DAB, \) and \( BDA \) based on relationships and supplementary angles in the circle.
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