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
Related questions
Question
![**Title: Understanding Inscribed Angles in Circles**
**Problem Statement:**
Triangle \(ELM\) is inscribed in a circle as shown. Find the measure of angle \(EMO\).
**Diagram Description:**
- The circle has a triangle \(ELM\) inscribed in it.
- Point \(L\) lies on the circle, forming an angle \(ELM\) which measures \(34^\circ\).
- Points \(E\), \(L\), and \(M\) lie on the circumference of the circle.
- Point \(O\) is the center of the circle.
- Line \(LM\) serves as a chord of the circle.
**Explanation:**
In this setup, we are tasked with finding angle \(EMO\). Since \(EMO\) is an angle subtended by the same arc \(LM\) as angle \(ELM\), which is inscribed in the circle:
- Using the Inscribed Angle Theorem, the angle subtended by an arc at the center of the circle (\(EMO\)) is twice the angle subtended at any point on the circle's circumference.
- Therefore, angle \(\angle EMO = 2 \times \angle ELM\).
- Given \(\angle ELM = 34^\circ\), calculate \(\angle EMO\):
\[
\angle EMO = 2 \times 34^\circ = 68^\circ
\]
Thus, the measure of angle \(EMO\) is \(68^\circ\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40ac1924-da82-497b-85f4-398551157699%2F865a2b73-0400-4db7-acf5-6861caf5243d%2Fllhbuhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Inscribed Angles in Circles**
**Problem Statement:**
Triangle \(ELM\) is inscribed in a circle as shown. Find the measure of angle \(EMO\).
**Diagram Description:**
- The circle has a triangle \(ELM\) inscribed in it.
- Point \(L\) lies on the circle, forming an angle \(ELM\) which measures \(34^\circ\).
- Points \(E\), \(L\), and \(M\) lie on the circumference of the circle.
- Point \(O\) is the center of the circle.
- Line \(LM\) serves as a chord of the circle.
**Explanation:**
In this setup, we are tasked with finding angle \(EMO\). Since \(EMO\) is an angle subtended by the same arc \(LM\) as angle \(ELM\), which is inscribed in the circle:
- Using the Inscribed Angle Theorem, the angle subtended by an arc at the center of the circle (\(EMO\)) is twice the angle subtended at any point on the circle's circumference.
- Therefore, angle \(\angle EMO = 2 \times \angle ELM\).
- Given \(\angle ELM = 34^\circ\), calculate \(\angle EMO\):
\[
\angle EMO = 2 \times 34^\circ = 68^\circ
\]
Thus, the measure of angle \(EMO\) is \(68^\circ\).
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