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
Problem 1CT
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![**Question:**
In circle \( F \) with \( m\angle EHB = 46^\circ \), find the angle measure of minor arc \( E \hat{G} \).
**Explanation of the Diagram:**
In the provided diagram on graph paper, a circle labeled \( F \) is depicted with three points on its circumference: \( E \), \( G \), and \( H \). Point \( E \) is connected to point \( H \) forming an angle \( EHB \) equal to \( 46^\circ \). The task involves finding the measure of the central angle subtended by the minor arc \( E \hat{G} \).
Here's how to approach this problem:
1. Identify the given angle \( m\angle EHB = 46^\circ \).
2. Recognize that the central angle \( \angle EFG \) subtended by minor arc \( E \hat{G} \) will be twice the measure of the inscribed angle \( \angle EHB \) according to the Inscribed Angle Theorem.
Given the Inscribed Angle Theorem:
\[ \text{Central Angle} = 2 \times \text{Inscribed Angle} \]
\[ m\angle EFG = 2 \times m\angle EHB \]
\[ m\angle EFG = 2 \times 46^\circ = 92^\circ \]
Thus, the measure of the minor arc \( E \hat{G} \) is \( 92^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61c86ddb-00aa-4043-ac36-ebc52fad487f%2Fbf26b189-fe14-4ef6-b3cd-d7962f27f69c%2Furc65lc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
In circle \( F \) with \( m\angle EHB = 46^\circ \), find the angle measure of minor arc \( E \hat{G} \).
**Explanation of the Diagram:**
In the provided diagram on graph paper, a circle labeled \( F \) is depicted with three points on its circumference: \( E \), \( G \), and \( H \). Point \( E \) is connected to point \( H \) forming an angle \( EHB \) equal to \( 46^\circ \). The task involves finding the measure of the central angle subtended by the minor arc \( E \hat{G} \).
Here's how to approach this problem:
1. Identify the given angle \( m\angle EHB = 46^\circ \).
2. Recognize that the central angle \( \angle EFG \) subtended by minor arc \( E \hat{G} \) will be twice the measure of the inscribed angle \( \angle EHB \) according to the Inscribed Angle Theorem.
Given the Inscribed Angle Theorem:
\[ \text{Central Angle} = 2 \times \text{Inscribed Angle} \]
\[ m\angle EFG = 2 \times m\angle EHB \]
\[ m\angle EFG = 2 \times 46^\circ = 92^\circ \]
Thus, the measure of the minor arc \( E \hat{G} \) is \( 92^\circ \).
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