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6. Find mSPQ. P 45° 50⁰ R S100 BiU Font Family - AA- A

6. Find mSPQ. P 45° 50⁰ R S100 BiU Font Family - AA- A

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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### Problem 6: Find m∠SPQ. #### Diagram Description: The diagram features a circle with points labeled P, Q, R, S, and T on its circumference, and a line segment UT extending tangentially from the circle at point S. The circle has arcs and angles marked as follows: - Arc PS is labeled as 45°. - Arc PQ is labeled as 50°. - Arc RS is labeled as 100°. Points P and Q are connected by line PQ, and other lines such as ST, PR, and UT are also depicted in the diagram. Point T is the intersection point inside the circle. To find m∠SPQ, observe the position and relationships of the provided angles and line segments: - Angle PST (angle formed by points P, S, and T along with circle arc PS) - Angle QRT (angle formed by points Q, R, and T along with circle arc QR) - Tangent line UT aligned with point S, indicating that angle SUT is right (90°). By analyzing the given arc measures and diagram relationships, mathematical principles such as circle theorems and tangent-segment angles can be applied to calculate the measure. #### Solution Breakdown: Analyzing the arc and angle relationships based on the circle's properties and leveraging the known angles will facilitate determination of angle SPQ. Use relevant geometric theorems involving intersecting chords, tangents, and arcs within the circle to determine the required angle measure. To accurately determine the precise measure of angle SPQ, consider: - Sum of angles around point T. - Angles subtended by the same arc. - Angle formed by intersection with tangents. Understanding these relationships will guide you towards the solution.
Transcribed Image Text:### Problem 6: Find m∠SPQ. #### Diagram Description: The diagram features a circle with points labeled P, Q, R, S, and T on its circumference, and a line segment UT extending tangentially from the circle at point S. The circle has arcs and angles marked as follows: - Arc PS is labeled as 45°. - Arc PQ is labeled as 50°. - Arc RS is labeled as 100°. Points P and Q are connected by line PQ, and other lines such as ST, PR, and UT are also depicted in the diagram. Point T is the intersection point inside the circle. To find m∠SPQ, observe the position and relationships of the provided angles and line segments: - Angle PST (angle formed by points P, S, and T along with circle arc PS) - Angle QRT (angle formed by points Q, R, and T along with circle arc QR) - Tangent line UT aligned with point S, indicating that angle SUT is right (90°). By analyzing the given arc measures and diagram relationships, mathematical principles such as circle theorems and tangent-segment angles can be applied to calculate the measure. #### Solution Breakdown: Analyzing the arc and angle relationships based on the circle's properties and leveraging the known angles will facilitate determination of angle SPQ. Use relevant geometric theorems involving intersecting chords, tangents, and arcs within the circle to determine the required angle measure. To accurately determine the precise measure of angle SPQ, consider: - Sum of angles around point T. - Angles subtended by the same arc. - Angle formed by intersection with tangents. Understanding these relationships will guide you towards the solution.
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