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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![**Educational Website Content**
### Determining Angles in a Circle
**Problem Statement:**
Given the circle where \( \overline{RS} \cong \overline{UT} \), determine the value of \( x \) and \( m\angle T\hat{U} \).
**Description of Diagram:**
In the given circle, there are two chords, \( \overline{RS} \) and \( \overline{UT} \), which are congruent (\( \overline{RS} \cong \overline{UT} \)). There are two central angles associated with these chords.
1. \( \angle R\hat{S} \) is labeled as \( (5x + 8)^\circ \).
2. \( \angle U\hat{T} \) is labeled as \( (3x + 14)^\circ \).
**Steps to Solve:**
1. **Recognizing Congruent Arcs:**
Since \( \overline{RS} \cong \overline{UT} \), the angles subtended by these chords at the center of the circle must also be equal:
\[
\angle R\hat{S} = \angle U\hat{T}.
\]
2. **Set Up the Equation:**
Given:
\[
5x + 8 = 3x + 14
\]
3. **Solve for \( x \):**
\[
5x + 8 = 3x + 14 \\
5x - 3x = 14 - 8 \\
2x = 6 \\
x = 3
\]
4. **Determine \( m\angle T\hat{U} \):**
Substitute \( x = 3 \) into either angle expression. Using \( \angle R\hat{S} = 5x + 8 \):
\[
m\angle T\hat{U} = 5(3) + 8 = 15 + 8 = 23^\circ.
\]
Therefore, \( x = 3 \) and \( m\angle T\hat{U} = 23^\circ \).
**Conclusion:**
By understanding the properties of congruent chords and their subtended angles in a circle, we determined](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa242a625-2826-4f00-ad8f-039422d3d327%2Fc0a162f5-ba30-4791-b458-112688b613f3%2Fmn8avxy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content**
### Determining Angles in a Circle
**Problem Statement:**
Given the circle where \( \overline{RS} \cong \overline{UT} \), determine the value of \( x \) and \( m\angle T\hat{U} \).
**Description of Diagram:**
In the given circle, there are two chords, \( \overline{RS} \) and \( \overline{UT} \), which are congruent (\( \overline{RS} \cong \overline{UT} \)). There are two central angles associated with these chords.
1. \( \angle R\hat{S} \) is labeled as \( (5x + 8)^\circ \).
2. \( \angle U\hat{T} \) is labeled as \( (3x + 14)^\circ \).
**Steps to Solve:**
1. **Recognizing Congruent Arcs:**
Since \( \overline{RS} \cong \overline{UT} \), the angles subtended by these chords at the center of the circle must also be equal:
\[
\angle R\hat{S} = \angle U\hat{T}.
\]
2. **Set Up the Equation:**
Given:
\[
5x + 8 = 3x + 14
\]
3. **Solve for \( x \):**
\[
5x + 8 = 3x + 14 \\
5x - 3x = 14 - 8 \\
2x = 6 \\
x = 3
\]
4. **Determine \( m\angle T\hat{U} \):**
Substitute \( x = 3 \) into either angle expression. Using \( \angle R\hat{S} = 5x + 8 \):
\[
m\angle T\hat{U} = 5(3) + 8 = 15 + 8 = 23^\circ.
\]
Therefore, \( x = 3 \) and \( m\angle T\hat{U} = 23^\circ \).
**Conclusion:**
By understanding the properties of congruent chords and their subtended angles in a circle, we determined
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