Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter4: Quadrilaterals
Section4.2: The Parallelogram And Kite
Problem 15E
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![### Understanding Angle Measures in a Cyclic Quadrilateral
#### Problem Statement:
**What is the value of \( x \)?**
#### Diagram Description:
The diagram shows a cyclic quadrilateral inscribed in a circle. The vertices of the quadrilateral are labeled \( G, F, H, \) and \( I \), with the points arranged consecutively along the circle. The specific angle measures provided in the diagram are:
- \( \angle GHF = 2x \)
- \( \angle FGI = x+18 \)
#### Explanation:
To find the value of \( x \), we should recall the properties of a cyclic quadrilateral:
1. **Opposite Angles of Cyclic Quadrilateral**: The opposite angles of a cyclic quadrilateral sum up to \( 180^\circ \).
Using this property:
Given the opposite angles \( \angle GHF \) and \( \angle FGI \):
\[ 2x + (x + 18) = 180^\circ \]
2. **Solving for \( x \)**:
Combine the terms and solve for \( x \):
\[ 3x + 18 = 180^\circ \]
\[ 3x = 162^\circ \]
\[ x = \frac{162^\circ}{3} \]
\[ x = 54^\circ \]
Thus, the value of \( x \) is:
\[ x = 54^\circ \]
This problem demonstrates the use of properties of cyclic quadrilaterals to solve for unknown angle measures, emphasizing the importance of understanding these geometric principles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F056e647e-6927-46de-b6aa-8f2eb320dc65%2Fa2e1893d-1ab8-45b0-9890-4d4518fa4362%2F3t6nlmh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Angle Measures in a Cyclic Quadrilateral
#### Problem Statement:
**What is the value of \( x \)?**
#### Diagram Description:
The diagram shows a cyclic quadrilateral inscribed in a circle. The vertices of the quadrilateral are labeled \( G, F, H, \) and \( I \), with the points arranged consecutively along the circle. The specific angle measures provided in the diagram are:
- \( \angle GHF = 2x \)
- \( \angle FGI = x+18 \)
#### Explanation:
To find the value of \( x \), we should recall the properties of a cyclic quadrilateral:
1. **Opposite Angles of Cyclic Quadrilateral**: The opposite angles of a cyclic quadrilateral sum up to \( 180^\circ \).
Using this property:
Given the opposite angles \( \angle GHF \) and \( \angle FGI \):
\[ 2x + (x + 18) = 180^\circ \]
2. **Solving for \( x \)**:
Combine the terms and solve for \( x \):
\[ 3x + 18 = 180^\circ \]
\[ 3x = 162^\circ \]
\[ x = \frac{162^\circ}{3} \]
\[ x = 54^\circ \]
Thus, the value of \( x \) is:
\[ x = 54^\circ \]
This problem demonstrates the use of properties of cyclic quadrilaterals to solve for unknown angle measures, emphasizing the importance of understanding these geometric principles.
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