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
![**Geometry Problem: Circle and Angles**
**Problem Statement:**
In circle \( Q \) with \( m\angle PSR = 30^\circ \), find the \( m\angle PQR \).
**Diagram Explanation:**
The diagram illustrates a circle with center \( Q \). Points \( P \), \( S \), and \( R \) lie on the circumference of the circle, and line segments \( PQ \), \( QS \), and \( QR \) are drawn. The angle \( \angle PSR \) is given as \( 30^\circ \).
**Solution:**
To find the measure of angle \( \angle PQR \):
1. Understand that in this problem, points \( P \), \( Q \), \( R \), and \( S \) lie on the circle with \( Q \) being the center.
2. The given angle \( \angle PSR = 30^\circ \) is an inscribed angle.
3. An inscribed angle in a circle is half the measure of the central angle that subtends the same arc.
4. The central angle, \( \angle PQR \), subtending the same arc as \( \angle PSR \) will therefore be twice the measure of \( \angle PSR \).
Thus:
\[ m\angle PQR = 2 \times m\angle PSR \]
\[ m\angle PQR = 2 \times 30^\circ \]
\[ m\angle PQR = 60^\circ \]
**Answer:**
\[ m\angle PQR = 60^\circ \]
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Transcribed Image Text:**Geometry Problem: Circle and Angles**
**Problem Statement:**
In circle \( Q \) with \( m\angle PSR = 30^\circ \), find the \( m\angle PQR \).
**Diagram Explanation:**
The diagram illustrates a circle with center \( Q \). Points \( P \), \( S \), and \( R \) lie on the circumference of the circle, and line segments \( PQ \), \( QS \), and \( QR \) are drawn. The angle \( \angle PSR \) is given as \( 30^\circ \).
**Solution:**
To find the measure of angle \( \angle PQR \):
1. Understand that in this problem, points \( P \), \( Q \), \( R \), and \( S \) lie on the circle with \( Q \) being the center.
2. The given angle \( \angle PSR = 30^\circ \) is an inscribed angle.
3. An inscribed angle in a circle is half the measure of the central angle that subtends the same arc.
4. The central angle, \( \angle PQR \), subtending the same arc as \( \angle PSR \) will therefore be twice the measure of \( \angle PSR \).
Thus:
\[ m\angle PQR = 2 \times m\angle PSR \]
\[ m\angle PQR = 2 \times 30^\circ \]
\[ m\angle PQR = 60^\circ \]
**Answer:**
\[ m\angle PQR = 60^\circ \]
**Submit your answer:**
[Answer Input Box]
[Submit Answer Button]
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