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
Related questions
Question
![### Transcription of Educational Content:
**Problem Statement:**
In circle H with \( m \angle GHJ = 66^\circ \), find the \( m \angle GKJ \).
**Graphical Representation:**
The image presents a circle labeled as circle H. Within this circle, there are four points labeled as G, H, J, and K. Here's a detailed description of the geometric configuration:
- Point G, Point H, Point J, and Point K are on the circumference of the circle.
- Line segment GH and line segment HJ are connected, forming the angle \( \angle GHJ \).
- Line segments GJ and JK intersect at point K on the circumference of the circle.
- A central point is not explicitly identified in the setup.
**Angle Relations in the Circle:**
- \( m \angle GHJ \) is given as \(66^\circ \).
- We are required to find the measure of angle \( \angle GKJ \).
**Analysis:**
In a cyclic quadrilateral (a four-sided figure where all vertices lie on the circumference of the circle), opposite angles are supplementary. Therefore, if \(\angle GHJ\) is \(66^\circ\), we can apply this property to find \( \angle GKJ \).
Since \( GHJK \) forms a cyclic quadrilateral:
\[
\angle GHJ + \angle GKJ = 180^\circ
\]
Thus, \( \angle GKJ \) can be found using the equation:
\[
\angle GKJ = 180^\circ - 66^\circ = 114^\circ
\]
So, \( m \angle GKJ = 114^\circ \).
---
This educational problem involves understanding properties of angles in a cyclic quadrilateral and applying these properties to find unknown angle measures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccc6c07f-901f-4b9b-9990-bf05e0ed87d5%2Fb404c484-622c-4920-ae0f-310050e729e2%2Fxr3kuh_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription of Educational Content:
**Problem Statement:**
In circle H with \( m \angle GHJ = 66^\circ \), find the \( m \angle GKJ \).
**Graphical Representation:**
The image presents a circle labeled as circle H. Within this circle, there are four points labeled as G, H, J, and K. Here's a detailed description of the geometric configuration:
- Point G, Point H, Point J, and Point K are on the circumference of the circle.
- Line segment GH and line segment HJ are connected, forming the angle \( \angle GHJ \).
- Line segments GJ and JK intersect at point K on the circumference of the circle.
- A central point is not explicitly identified in the setup.
**Angle Relations in the Circle:**
- \( m \angle GHJ \) is given as \(66^\circ \).
- We are required to find the measure of angle \( \angle GKJ \).
**Analysis:**
In a cyclic quadrilateral (a four-sided figure where all vertices lie on the circumference of the circle), opposite angles are supplementary. Therefore, if \(\angle GHJ\) is \(66^\circ\), we can apply this property to find \( \angle GKJ \).
Since \( GHJK \) forms a cyclic quadrilateral:
\[
\angle GHJ + \angle GKJ = 180^\circ
\]
Thus, \( \angle GKJ \) can be found using the equation:
\[
\angle GKJ = 180^\circ - 66^\circ = 114^\circ
\]
So, \( m \angle GKJ = 114^\circ \).
---
This educational problem involves understanding properties of angles in a cyclic quadrilateral and applying these properties to find unknown angle measures.
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