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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![### Geometry Problem Involving Angle Bisectors
In this geometry problem, we are given a diagram with three points: \( A \), \( B \), \( C \), and \( O \). The ray \( \overrightarrow{OB} \) bisects \( \angle AOC \). The measures of \( \angle AOB \) and \( \angle BOC \) are given as \( (11x - 3)^\circ \) and \( (7x + 9)^\circ \) respectively. Our goal is to find the measure of \( \angle AOC \).
### Diagram Explanation:
- Point \( O \) is the vertex of the angles.
- Ray \( \overrightarrow{OA} \), \( \overrightarrow{OB} \), and \( \overrightarrow{OC} \) are drawn from point \( O \).
- Point \( A \) lies on ray \( \overrightarrow{OA} \).
- Point \( B \) lies on ray \( \overrightarrow{OB} \).
- Point \( C \) lies on ray \( \overrightarrow{OC} \).
- The measure \( \angle AOB = (11x - 3)^\circ \).
- The measure \( \angle BOC = (7x + 9)^\circ \).
### Given Information:
- Ray \( \overrightarrow{OB} \) bisects \( \angle AOC \).
- \( \angle AOB = (11x - 3)^\circ \)
- \( \angle BOC = (7x + 9)^\circ \)
### Objective:
To find the measure of \( \angle AOC \).
### Solution:
1. Because ray \( \overrightarrow{OB} \) bisects \( \angle AOC \), we know that:
\[
\angle AOB = \angle BOC
\]
Therefore,
\[
11x - 3 = 7x + 9
\]
2. Solve for \( x \):
\[
11x - 7x = 9 + 3
\]
\[
4x = 12
\]
\[
x = 3
\]
3. Now substitute \( x = 3 \) back into the expressions for the angles to find their measures:
- \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c239b2-49f2-4118-83c1-4fc31b3c935a%2Fd8c79aae-c0bf-4fee-834f-ea4a47394037%2Fwytlc6e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem Involving Angle Bisectors
In this geometry problem, we are given a diagram with three points: \( A \), \( B \), \( C \), and \( O \). The ray \( \overrightarrow{OB} \) bisects \( \angle AOC \). The measures of \( \angle AOB \) and \( \angle BOC \) are given as \( (11x - 3)^\circ \) and \( (7x + 9)^\circ \) respectively. Our goal is to find the measure of \( \angle AOC \).
### Diagram Explanation:
- Point \( O \) is the vertex of the angles.
- Ray \( \overrightarrow{OA} \), \( \overrightarrow{OB} \), and \( \overrightarrow{OC} \) are drawn from point \( O \).
- Point \( A \) lies on ray \( \overrightarrow{OA} \).
- Point \( B \) lies on ray \( \overrightarrow{OB} \).
- Point \( C \) lies on ray \( \overrightarrow{OC} \).
- The measure \( \angle AOB = (11x - 3)^\circ \).
- The measure \( \angle BOC = (7x + 9)^\circ \).
### Given Information:
- Ray \( \overrightarrow{OB} \) bisects \( \angle AOC \).
- \( \angle AOB = (11x - 3)^\circ \)
- \( \angle BOC = (7x + 9)^\circ \)
### Objective:
To find the measure of \( \angle AOC \).
### Solution:
1. Because ray \( \overrightarrow{OB} \) bisects \( \angle AOC \), we know that:
\[
\angle AOB = \angle BOC
\]
Therefore,
\[
11x - 3 = 7x + 9
\]
2. Solve for \( x \):
\[
11x - 7x = 9 + 3
\]
\[
4x = 12
\]
\[
x = 3
\]
3. Now substitute \( x = 3 \) back into the expressions for the angles to find their measures:
- \( \
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