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
![### Geometry Angle Bisector Problem
#### Problem Statement
In the diagram, \( \overline{DC} \) bisects \( \angle BDA \). Find \( m \angle BDC \).
#### Diagram Explanation
The diagram features four points: A, B, C, and D, with the following angles:
- \( \angle BDA \) is bisected by \( \overline{DC} \).
- \( \angle BDC = 3x + 37 \) degrees.
- \( \angle CDA = 5x + 45 \) degrees.
The options provided for the value of \( m \angle BDC \) are:
1. 41°
2. 25°
3. 12.5°
4. 50°
You should use the angle bisector information along with the given angle expressions to solve the problem.
#### Solution
Given that \( \overline{DC} \) is the bisector, we know that:
\[ \angle BDC = \angle CDA \]
Thus,
\[ 3x + 37 = 5x + 45 \]
#### Solving for \( x \):
\[ 3x + 37 = 5x + 45 \]
\[ 37 - 45 = 5x - 3x \]
\[ -8 = 2x \]
\[ x = -4 \]
#### Calculating \( m \angle BDC \):
Substitute \( x = -4 \) back into the expression for \( \angle BDC \):
\[ \angle BDC = 3(-4) + 37 \]
\[ \angle BDC = -12 + 37 \]
\[ \angle BDC = 25° \]
Thus, the measure of \( \angle BDC \) is:
\[ 25° \]
### Answer: 25°
### Conclusion:
When \( \overline{DC} \) bisects \( \angle BDA \), the measure of \( \angle BDC \) is 25°.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d44061d-5891-4c6f-9f7e-bc7f6369bc06%2Fb9572aef-f0ac-45a7-889d-b05e4e80a5fb%2F35lawtv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Angle Bisector Problem
#### Problem Statement
In the diagram, \( \overline{DC} \) bisects \( \angle BDA \). Find \( m \angle BDC \).
#### Diagram Explanation
The diagram features four points: A, B, C, and D, with the following angles:
- \( \angle BDA \) is bisected by \( \overline{DC} \).
- \( \angle BDC = 3x + 37 \) degrees.
- \( \angle CDA = 5x + 45 \) degrees.
The options provided for the value of \( m \angle BDC \) are:
1. 41°
2. 25°
3. 12.5°
4. 50°
You should use the angle bisector information along with the given angle expressions to solve the problem.
#### Solution
Given that \( \overline{DC} \) is the bisector, we know that:
\[ \angle BDC = \angle CDA \]
Thus,
\[ 3x + 37 = 5x + 45 \]
#### Solving for \( x \):
\[ 3x + 37 = 5x + 45 \]
\[ 37 - 45 = 5x - 3x \]
\[ -8 = 2x \]
\[ x = -4 \]
#### Calculating \( m \angle BDC \):
Substitute \( x = -4 \) back into the expression for \( \angle BDC \):
\[ \angle BDC = 3(-4) + 37 \]
\[ \angle BDC = -12 + 37 \]
\[ \angle BDC = 25° \]
Thus, the measure of \( \angle BDC \) is:
\[ 25° \]
### Answer: 25°
### Conclusion:
When \( \overline{DC} \) bisects \( \angle BDA \), the measure of \( \angle BDC \) is 25°.
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