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
explain reasoning
![### Determining Angle 'a' in Figure 10.13
#### Problem Statement:
2. **Determine Angle 'a' in Figure 10.13. Explain your reasoning.**
#### Figure 10.13 Description:
The diagram consists of two parallel lines intersected by a transversal line and an additional non-parallel line.
- The parallel lines are indicated with arrows showing their parallel property.
- The transversal intersects both parallel lines.
- Angle measures provided in the diagram are 95° and 35°.
- Angle 'a' is located between the parallel lines and is created by the intersection of the transversal and another non-parallel line.
#### Explanation:
To find angle 'a':
1. **Identify Known Angles Relative to Parallel Lines**:
- The given angle of 95° adjacent to angle 'a' is an exterior angle to the interior angles formed by the intersection of the transversal with the non-parallel and parallel lines.
2. **Use Alternate Interior Angles Theorem**:
- Since parallel lines are intersected by a transversal, the alternate interior angles theorem applies. The angle adjacent to 'a' (adjacent to the intersection at the topmost parallel) will be equal to the angle located at the bottom intersection (35°).
- Angle 'a' is supplementary to the angle next to the 95° angle, because they form a linear pair along the line not parallel to others.
3. **Calculate Angle 'a'**:
- Using the supplementary angle rule:
- Let angle adjacent to 'a' be \( x \).
- Since these two angles are on a straight line, they sum up to \( 180^\circ \):
\[
x + 95^\circ = 180^\circ
\]
\[
x = 180^\circ - 95^\circ
\]
\[
x = 85^\circ
\]
- Therefore, \( a \) is supplementary to \( x \) which implies:
\[
a \) = 85^\circ
\]
Thus, \( a = 55^\circ \).
#### Conclusion:
Angle 'a' in Figure 10.13 is \( 85^\circ \), explained through the use of alternate interior angles theorem and supplementary angle properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c60e9dc-9541-469d-8098-00c54f69ea68%2F3a6840eb-1edb-4a7b-a677-b907c5be60e7%2Ffm0pufd_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining Angle 'a' in Figure 10.13
#### Problem Statement:
2. **Determine Angle 'a' in Figure 10.13. Explain your reasoning.**
#### Figure 10.13 Description:
The diagram consists of two parallel lines intersected by a transversal line and an additional non-parallel line.
- The parallel lines are indicated with arrows showing their parallel property.
- The transversal intersects both parallel lines.
- Angle measures provided in the diagram are 95° and 35°.
- Angle 'a' is located between the parallel lines and is created by the intersection of the transversal and another non-parallel line.
#### Explanation:
To find angle 'a':
1. **Identify Known Angles Relative to Parallel Lines**:
- The given angle of 95° adjacent to angle 'a' is an exterior angle to the interior angles formed by the intersection of the transversal with the non-parallel and parallel lines.
2. **Use Alternate Interior Angles Theorem**:
- Since parallel lines are intersected by a transversal, the alternate interior angles theorem applies. The angle adjacent to 'a' (adjacent to the intersection at the topmost parallel) will be equal to the angle located at the bottom intersection (35°).
- Angle 'a' is supplementary to the angle next to the 95° angle, because they form a linear pair along the line not parallel to others.
3. **Calculate Angle 'a'**:
- Using the supplementary angle rule:
- Let angle adjacent to 'a' be \( x \).
- Since these two angles are on a straight line, they sum up to \( 180^\circ \):
\[
x + 95^\circ = 180^\circ
\]
\[
x = 180^\circ - 95^\circ
\]
\[
x = 85^\circ
\]
- Therefore, \( a \) is supplementary to \( x \) which implies:
\[
a \) = 85^\circ
\]
Thus, \( a = 55^\circ \).
#### Conclusion:
Angle 'a' in Figure 10.13 is \( 85^\circ \), explained through the use of alternate interior angles theorem and supplementary angle properties.
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