75° R 95° ?

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
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What is the measure of arc SQ?

In this diagram, we have a circle with four points on its circumference labeled Q, R, S, and another point needing an angle measurement.

### Diagram Description:

1. **Chord RS** subtends an angle of 75° at the circumference on the opposite side from the center.
2. There is another angle, ∠SQR, formed by the intersection of the chords SQ and QR. This angle measures 95°.
3. An unknown angle at point Q is noted as "?" indicating that this angle measurement needs to be determined.

### Detailed Explanation:

The circle is divided by the chords into various segments and angles. The key relationships between these angles can be used to determine unknown measures.

#### Key Concepts:
- **Angles in the Same Segment**: Angles subtended by the same arc at the circumference are equal.
- **Cyclic Quadrilateral Property**: The opposite angles of a quadrilateral that is inscribed in a circle (cyclic quadrilateral) sum up to 180°.

Considering the cyclic quadrilateral QRSQ:
- Opposite angles of a cyclic quadrilateral sum to 180°.
- Given ∠QRS = 75°, and ∠SQR = 95°.

Then, the unknown angle will be:
- Calculate ∠QSR using the properties of cyclic quadrilaterals: The sum of ∠QRS and ∠QSR should equal 180°.
- ∠QSR = 180° - 75° = 105°

Hence, the unknown angle at point Q is 105°. 

This diagram visually represents the stated angle relationships and their properties in the context of a circle and helps in understanding geometrical principles related to cyclic quadrilaterals.
Transcribed Image Text:In this diagram, we have a circle with four points on its circumference labeled Q, R, S, and another point needing an angle measurement. ### Diagram Description: 1. **Chord RS** subtends an angle of 75° at the circumference on the opposite side from the center. 2. There is another angle, ∠SQR, formed by the intersection of the chords SQ and QR. This angle measures 95°. 3. An unknown angle at point Q is noted as "?" indicating that this angle measurement needs to be determined. ### Detailed Explanation: The circle is divided by the chords into various segments and angles. The key relationships between these angles can be used to determine unknown measures. #### Key Concepts: - **Angles in the Same Segment**: Angles subtended by the same arc at the circumference are equal. - **Cyclic Quadrilateral Property**: The opposite angles of a quadrilateral that is inscribed in a circle (cyclic quadrilateral) sum up to 180°. Considering the cyclic quadrilateral QRSQ: - Opposite angles of a cyclic quadrilateral sum to 180°. - Given ∠QRS = 75°, and ∠SQR = 95°. Then, the unknown angle will be: - Calculate ∠QSR using the properties of cyclic quadrilaterals: The sum of ∠QRS and ∠QSR should equal 180°. - ∠QSR = 180° - 75° = 105° Hence, the unknown angle at point Q is 105°. This diagram visually represents the stated angle relationships and their properties in the context of a circle and helps in understanding geometrical principles related to cyclic quadrilaterals.
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