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What is the value of x? X+47⁰ 60⁰ x-14° Q x+63° S

What is the value of x? X+47⁰ 60⁰ x-14° Q x+63° S

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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--- **Understanding Angles in Circles** **Problem: What is the value of \( x \)?** In the diagram below, we're given a circle with center \( Q \) and several angles. The goal is to find the value of \( x \). **Diagram Explanation:** The circle has four angles formed by lines intersecting at point \( Q \). These angles are: - One \( 60^\circ \) angle. - An angle labeled \( x + 47^\circ \). - An angle labeled \( x - 14^\circ \). - An angle labeled \( x + 63^\circ \). **Points on the Circle:** - \( T \) - \( S \) - \( R \) - \( U \) Point \( Q \) is the center of the circle. **Steps to Solve:** 1. **Understand that the angles around point \( Q \) must sum up to \( 360^\circ \).** 2. **Set up the equation:** \( 60^\circ + (x + 47^\circ) + (x - 14^\circ) + (x + 63^\circ) = 360^\circ \) Simplify the equation: \[ 60 + x + 47 + x - 14 + x + 63 = 360 \] Combine like terms: \[ 3x + 156 = 360 \] Solve for \( x \): \[ 3x = 360 - 156 \] \[ 3x = 204 \] \[ x = 68 \] **Answer: \( x = 68^\circ \)** --- This completes the problem-solving process for finding the value of \( x \) based on the angles provided in the circle.
Transcribed Image Text:--- **Understanding Angles in Circles** **Problem: What is the value of \( x \)?** In the diagram below, we're given a circle with center \( Q \) and several angles. The goal is to find the value of \( x \). **Diagram Explanation:** The circle has four angles formed by lines intersecting at point \( Q \). These angles are: - One \( 60^\circ \) angle. - An angle labeled \( x + 47^\circ \). - An angle labeled \( x - 14^\circ \). - An angle labeled \( x + 63^\circ \). **Points on the Circle:** - \( T \) - \( S \) - \( R \) - \( U \) Point \( Q \) is the center of the circle. **Steps to Solve:** 1. **Understand that the angles around point \( Q \) must sum up to \( 360^\circ \).** 2. **Set up the equation:** \( 60^\circ + (x + 47^\circ) + (x - 14^\circ) + (x + 63^\circ) = 360^\circ \) Simplify the equation: \[ 60 + x + 47 + x - 14 + x + 63 = 360 \] Combine like terms: \[ 3x + 156 = 360 \] Solve for \( x \): \[ 3x = 360 - 156 \] \[ 3x = 204 \] \[ x = 68 \] **Answer: \( x = 68^\circ \)** --- This completes the problem-solving process for finding the value of \( x \) based on the angles provided in the circle.
### Solving the Measure of Angle ∠UTW To solve for the measure of angle ∠UTW within a circle, we start by understanding that the measures of the angles ∠UTV, ∠UTW, and ∠VTW sum to 360° because they span an entire circle. #### Step-by-Step Solution 1. **Write the Equation**: \[ m∠UTV + m∠UTW + m∠VTW = 360° \] 2. **Substitute Known Values**: \[ 130° + m∠UTW + 90° = 360° \] - Here, \( m∠UTV = 130° \) and \( m∠VTW = 90° \). 3. **Combine Like Terms**: \[ m∠UTW + 220° = 360° \] 4. **Isolate \( m∠UTW \)**: \[ m∠UTW = 360° - 220° \] 5. **Solve the Equation**: \[ m∠UTW = 140° \] Thus, the measure of angle ∠UTW is \( 140° \). ### Summary We have determined that \( m∠UTW \) is \( 140° \) by formulating the equation based on the sum of angles in a circle, substituting the known values, combining like terms, and isolating the unknown angle. This method ensures a comprehensive understanding of how the measures of angles that span a circle relate to each other.
Transcribed Image Text:### Solving the Measure of Angle ∠UTW To solve for the measure of angle ∠UTW within a circle, we start by understanding that the measures of the angles ∠UTV, ∠UTW, and ∠VTW sum to 360° because they span an entire circle. #### Step-by-Step Solution 1. **Write the Equation**: \[ m∠UTV + m∠UTW + m∠VTW = 360° \] 2. **Substitute Known Values**: \[ 130° + m∠UTW + 90° = 360° \] - Here, \( m∠UTV = 130° \) and \( m∠VTW = 90° \). 3. **Combine Like Terms**: \[ m∠UTW + 220° = 360° \] 4. **Isolate \( m∠UTW \)**: \[ m∠UTW = 360° - 220° \] 5. **Solve the Equation**: \[ m∠UTW = 140° \] Thus, the measure of angle ∠UTW is \( 140° \). ### Summary We have determined that \( m∠UTW \) is \( 140° \) by formulating the equation based on the sum of angles in a circle, substituting the known values, combining like terms, and isolating the unknown angle. This method ensures a comprehensive understanding of how the measures of angles that span a circle relate to each other.
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