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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![### Measuring Arc Lengths in Circles
**Problem:** Find the measure of arc WX.
**Diagram Explanation:**
The diagram consists of a circle with three points labeled W, X, and Y on its circumference. Two angles are marked inside the circle, one at 75° between points W, X, and another point. The other angle is marked as 110° between points Y, X, and another point.
### Concepts and Steps:
1. **Understanding the Circle:**
- A circle has 360 degrees.
- Arcs in a circle can be measured in degrees, which is a part of the total 360 degrees of the entire circle.
2. **Angles and Arcs Relationship:**
- The angle at the center of the circle is called the central angle.
- The measure of an arc is equal to the measure of its central angle.
3. **Calculation:**
- The angles provided are inside the circle.
- To find the measure of arc WX, observe that it is opposite to an angle provided.
- The sum of the angles in a circle around a point is 360°.
### Solution:
Given angles ∠YXW = 75° and ∠XWY = 110°,
- These angles are inside the circle, and together with the angles we are going to find.
To find the measure of arc WX, subtract from 360°:
\[ 360° - (75° + 110°) = 360° - 185° = 175° \]
Thus, the measure of arc WX is 175°.
### Conclusion:
The measure of arc WX is 175 degrees.
This method of measuring arc lengths is fundamental in geometry, specifically in topics involving circles and their properties. Understanding how to find arc lengths using given angles is crucial for solving more complex geometric problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2994886d-ee4f-443d-b178-240cdf3d6c71%2F708348c6-f52b-4e2c-940c-42a31349badb%2Fkat23gl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Measuring Arc Lengths in Circles
**Problem:** Find the measure of arc WX.
**Diagram Explanation:**
The diagram consists of a circle with three points labeled W, X, and Y on its circumference. Two angles are marked inside the circle, one at 75° between points W, X, and another point. The other angle is marked as 110° between points Y, X, and another point.
### Concepts and Steps:
1. **Understanding the Circle:**
- A circle has 360 degrees.
- Arcs in a circle can be measured in degrees, which is a part of the total 360 degrees of the entire circle.
2. **Angles and Arcs Relationship:**
- The angle at the center of the circle is called the central angle.
- The measure of an arc is equal to the measure of its central angle.
3. **Calculation:**
- The angles provided are inside the circle.
- To find the measure of arc WX, observe that it is opposite to an angle provided.
- The sum of the angles in a circle around a point is 360°.
### Solution:
Given angles ∠YXW = 75° and ∠XWY = 110°,
- These angles are inside the circle, and together with the angles we are going to find.
To find the measure of arc WX, subtract from 360°:
\[ 360° - (75° + 110°) = 360° - 185° = 175° \]
Thus, the measure of arc WX is 175°.
### Conclusion:
The measure of arc WX is 175 degrees.
This method of measuring arc lengths is fundamental in geometry, specifically in topics involving circles and their properties. Understanding how to find arc lengths using given angles is crucial for solving more complex geometric problems.
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