Find HLK. Show your work.

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
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Find HLK. Show your work. 

# Educational Analysis of Angles in a Circle

### Problem Statement:
Given a circle with points \(H\), \(L\), \(K\), \(M\), \(I\), and \(J\) positioned as shown in the diagram, where:
- \(m \overset{\frown}{JI} = (3x + 2)^\circ\)
- \(m \overset{\frown}{HLK} = (15x - 36)^\circ\)
- \(m \angle HMK = (8x - 1)^\circ\)

Determine the measure of \(m \overset{\frown}{HLK}\).

### Detailed Diagram Explanation:
The diagram illustrates a circle with several labeled points \(H\), \(L\), \(K\), \(M\), \(I\), and \(J\). The points and lines are arranged as follows:
- \(H\), \(J\), and \(I\) lie on the circumference.
- \(L\), \(K\), and \(M\) also lie on the circumference.
- \(M\) is a common intersection point for lines \(HJ\) and \(KL\).

### Solution:
To find \(m \overset{\frown}{HLK}\), we should use the relationships between the given angles and the segments of the circle.

1. **Identify Relationships Between the Given Angles:**
   - \(m \overset{\frown}{JI} = (3x + 2)^\circ\)
   - \(m \overset{\frown}{HLK} = (15x - 36)^\circ\)
   - \(m \angle HMK = (8x - 1)^\circ\)

2. **Use Circle Angle Properties:**
   - The sum of the central angles in a circle is 360 degrees.
   - The angle \( \angle HMK \) could be indirectly related to arcs \( \overset{\frown}{JI} \) and \( \overset{\frown}{HLK} \).

Solving the provided expressions will yield the answer.

## Solving the Equations:

### Step 1: Set Up the Equation
The angle \( \angle HMK \) relates to the arc measures it intercepts:
\[ m \angle HMK = \frac{1}{2} \left( m \overset{\frown
Transcribed Image Text:# Educational Analysis of Angles in a Circle ### Problem Statement: Given a circle with points \(H\), \(L\), \(K\), \(M\), \(I\), and \(J\) positioned as shown in the diagram, where: - \(m \overset{\frown}{JI} = (3x + 2)^\circ\) - \(m \overset{\frown}{HLK} = (15x - 36)^\circ\) - \(m \angle HMK = (8x - 1)^\circ\) Determine the measure of \(m \overset{\frown}{HLK}\). ### Detailed Diagram Explanation: The diagram illustrates a circle with several labeled points \(H\), \(L\), \(K\), \(M\), \(I\), and \(J\). The points and lines are arranged as follows: - \(H\), \(J\), and \(I\) lie on the circumference. - \(L\), \(K\), and \(M\) also lie on the circumference. - \(M\) is a common intersection point for lines \(HJ\) and \(KL\). ### Solution: To find \(m \overset{\frown}{HLK}\), we should use the relationships between the given angles and the segments of the circle. 1. **Identify Relationships Between the Given Angles:** - \(m \overset{\frown}{JI} = (3x + 2)^\circ\) - \(m \overset{\frown}{HLK} = (15x - 36)^\circ\) - \(m \angle HMK = (8x - 1)^\circ\) 2. **Use Circle Angle Properties:** - The sum of the central angles in a circle is 360 degrees. - The angle \( \angle HMK \) could be indirectly related to arcs \( \overset{\frown}{JI} \) and \( \overset{\frown}{HLK} \). Solving the provided expressions will yield the answer. ## Solving the Equations: ### Step 1: Set Up the Equation The angle \( \angle HMK \) relates to the arc measures it intercepts: \[ m \angle HMK = \frac{1}{2} \left( m \overset{\frown
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