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In circle Q with mZPSR = 30, find the mZPQR. S Q P. R

In circle Q with mZPSR = 30, find the mZPQR. S Q P. R

Elementary Geometry For College Students, 7e
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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### Geometry Problem - Circle Q #### Problem Statement: In circle \( Q \) with \( m\angle PSR = 30^\circ \), find the \( m\angle PQR \). #### Diagram Explanation: The given diagram includes: - A circle centered at point \( Q \). - Points \( P \), \( S \), and \( R \) lie on the circumference of the circle. - Line segment \( PS \) intersects \( SR \) at point \( S \). - Line segments \( PQ \) and \( QR \) connect points \( P \) and \( R \) to the center \( Q \) of the circle. #### Graph or Diagram Explanation: 1. **Circle Q:** A circle is shown with point \( Q \) at its center. 2. **Points on the Circumference:** Points \( P \), \( S \), and \( R \) are located on the circumference of the circle. 3. **Segments and Angles:** - Segment \( PS \) from \( P \) to \( S \). - Segment \( SR \) from \( S \) to \( R \). - Angle \( \angle PSR \) is marked as \( 30^\circ \). - Segment \( PQ \) from \( P \) to the center \( Q \). - Segment \( QR \) from \( Q \) to \( R \). - The task is to find the measure of angle \( \angle PQR \). #### Interactive Element: - **Answer Box:** An input box is provided for students to submit their answer. - **Submit Answer Button:** A blue button labeled "Submit Answer" allows students to submit their response. #### Solution Explanation: 1. Recognize that \( \angle PSR \) is an inscribed angle, and by the Inscribed Angle Theorem, \( \angle PSR \) subtends an arc \( PR \). 2. The measure of the arc \( PR \) is twice the inscribed angle, so \( \text{arc } PR = 2 \times 30^\circ = 60^\circ \). 3. Realize that \( \angle PQR \) is a central angle subtending the same arc \( PR \). 4. So, the measure of \( \angle PQR \) is equal to the measure of the arc it subtends. Thus, \( m\angle PQR
Transcribed Image Text:### Geometry Problem - Circle Q #### Problem Statement: In circle \( Q \) with \( m\angle PSR = 30^\circ \), find the \( m\angle PQR \). #### Diagram Explanation: The given diagram includes: - A circle centered at point \( Q \). - Points \( P \), \( S \), and \( R \) lie on the circumference of the circle. - Line segment \( PS \) intersects \( SR \) at point \( S \). - Line segments \( PQ \) and \( QR \) connect points \( P \) and \( R \) to the center \( Q \) of the circle. #### Graph or Diagram Explanation: 1. **Circle Q:** A circle is shown with point \( Q \) at its center. 2. **Points on the Circumference:** Points \( P \), \( S \), and \( R \) are located on the circumference of the circle. 3. **Segments and Angles:** - Segment \( PS \) from \( P \) to \( S \). - Segment \( SR \) from \( S \) to \( R \). - Angle \( \angle PSR \) is marked as \( 30^\circ \). - Segment \( PQ \) from \( P \) to the center \( Q \). - Segment \( QR \) from \( Q \) to \( R \). - The task is to find the measure of angle \( \angle PQR \). #### Interactive Element: - **Answer Box:** An input box is provided for students to submit their answer. - **Submit Answer Button:** A blue button labeled "Submit Answer" allows students to submit their response. #### Solution Explanation: 1. Recognize that \( \angle PSR \) is an inscribed angle, and by the Inscribed Angle Theorem, \( \angle PSR \) subtends an arc \( PR \). 2. The measure of the arc \( PR \) is twice the inscribed angle, so \( \text{arc } PR = 2 \times 30^\circ = 60^\circ \). 3. Realize that \( \angle PQR \) is a central angle subtending the same arc \( PR \). 4. So, the measure of \( \angle PQR \) is equal to the measure of the arc it subtends. Thus, \( m\angle PQR
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