Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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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.
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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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