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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![**Title: Determining the Value of an Unknown Angle in a Circle**
**Introduction:**
In this exercise, we will determine the value of \( x \) by analyzing the given angles in a circular diagram. The angles provided are part of a circle's interior, summing up to 360 degrees. This task involves basic geometric principles and methodical calculations.
**Diagram Description:**
The given diagram is a circle with center labelled, and four internal angles formed by lines drawn from the center to the circle's circumference at points \( T \), \( U \), \( V \), and \( S \). The angles are labeled as follows:
- Angle \( \angle STV = 96^\circ \)
- Angle \( \angle TUV = 60^\circ \)
- Angle \( \angle SUV = 75^\circ \)
- Angle \( \angle TUV = 3x \)
**Content:**
Here are the steps involved in solving for the unknown angle \( x \):
1. **Identify Known Values and Sum of Angles:**
- The sum of all interior angles around the center of a circle equals \( 360^\circ \).
2. **Given Angles:**
- \( \angle STV = 96^\circ \)
- \( \angle TUV = 60^\circ \)
- \( \angle SUS = 75^\circ \)
- The unknown angle is represented as \( \angle VUS = 3x \)
3. **Set Up the Equation:**
- The formula to find the unknown angle is:
\[ \text{Sum of given angles} + \text{Unknown Angle} = 360^\circ \]
\[ 96^\circ + 60^\circ + 75^\circ + 3x = 360^\circ \]
4. **Simplify and Solve for \( x \):**
- Combine like terms on the left side:
\[ 231^\circ + 3x = 360^\circ \]
- Isolate \( x \) by subtracting \( 231^\circ \) from both sides:
\[ 3x = 129^\circ \]
- Divide by 3:
\[ x = \frac{129^\circ}{3} = 43^\circ \]
**Conclusion:**
By analyzing the angles within the circle and using the sum of angles](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f6754e4-b4a1-4709-8cf0-90421fcab68c%2F35afe7a2-6d72-4c8c-b631-41ba9f472ae3%2Fpzb01v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determining the Value of an Unknown Angle in a Circle**
**Introduction:**
In this exercise, we will determine the value of \( x \) by analyzing the given angles in a circular diagram. The angles provided are part of a circle's interior, summing up to 360 degrees. This task involves basic geometric principles and methodical calculations.
**Diagram Description:**
The given diagram is a circle with center labelled, and four internal angles formed by lines drawn from the center to the circle's circumference at points \( T \), \( U \), \( V \), and \( S \). The angles are labeled as follows:
- Angle \( \angle STV = 96^\circ \)
- Angle \( \angle TUV = 60^\circ \)
- Angle \( \angle SUV = 75^\circ \)
- Angle \( \angle TUV = 3x \)
**Content:**
Here are the steps involved in solving for the unknown angle \( x \):
1. **Identify Known Values and Sum of Angles:**
- The sum of all interior angles around the center of a circle equals \( 360^\circ \).
2. **Given Angles:**
- \( \angle STV = 96^\circ \)
- \( \angle TUV = 60^\circ \)
- \( \angle SUS = 75^\circ \)
- The unknown angle is represented as \( \angle VUS = 3x \)
3. **Set Up the Equation:**
- The formula to find the unknown angle is:
\[ \text{Sum of given angles} + \text{Unknown Angle} = 360^\circ \]
\[ 96^\circ + 60^\circ + 75^\circ + 3x = 360^\circ \]
4. **Simplify and Solve for \( x \):**
- Combine like terms on the left side:
\[ 231^\circ + 3x = 360^\circ \]
- Isolate \( x \) by subtracting \( 231^\circ \) from both sides:
\[ 3x = 129^\circ \]
- Divide by 3:
\[ x = \frac{129^\circ}{3} = 43^\circ \]
**Conclusion:**
By analyzing the angles within the circle and using the sum of angles
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