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
![### Understanding Transversal Angles
#### Problem Statement
*Use the figure to answer the following question when \(a \parallel b\)*:
![Transversal Diagram]
If \( m \angle 1 = 80^\circ \), find \( m \angle 8 \).
#### Diagram Explanation
The provided diagram shows two parallel lines \(a\) and \(b\) that are intersected by a transversal line \(t\). This creates eight angles at the intersections. The angles are labeled 1 through 8. The goal is to find the measure of angle 8 given that the measure of angle 1 is 80 degrees.
Here is a step-by-step explanation for solving this problem:
1. **Identify the given information**:
- \( a \parallel b \)
- \( m \angle 1 = 80^\circ \)
2. **Understand the relationships**:
- Vertically opposite angles are equal. Therefore, \( m \angle 1 = m \angle 4 \) and \( m \angle 5 = m \angle 8 \).
3. **Recognize parallel lines with a transversal**:
- Alternate interior angles are equal. Therefore, \( m \angle 1 = m \angle 6 \) and \( m \angle 4 = m \angle 5 \).
- Corresponding angles are equal. Therefore, \( m \angle 1 = m \angle 5 \) and \( m \angle 4 = m \angle 8 \).
4. **Calculating other angles**:
- Since \( m \angle 1 = 80^\circ \), by corresponding angles, \( m \angle 5 = 80^\circ \).
- Since \( m \angle 5 = 80^\circ \) and vertically opposite angles are equal, \( m \angle 8 = 80^\circ \).
#### Conclusion
Therefore, \( m \angle 8 = 80^\circ \).
When a transversal intersects two parallel lines, it creates equal corresponding and alternate interior angles. This understanding helps to determine the measure of unknown angles based on known measures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20047395-754f-4b4f-b55f-051358973abe%2F0c3c9315-cd7a-41de-9737-d9b441f9d8b9%2Ftulwy2.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Transversal Angles
#### Problem Statement
*Use the figure to answer the following question when \(a \parallel b\)*:
![Transversal Diagram]
If \( m \angle 1 = 80^\circ \), find \( m \angle 8 \).
#### Diagram Explanation
The provided diagram shows two parallel lines \(a\) and \(b\) that are intersected by a transversal line \(t\). This creates eight angles at the intersections. The angles are labeled 1 through 8. The goal is to find the measure of angle 8 given that the measure of angle 1 is 80 degrees.
Here is a step-by-step explanation for solving this problem:
1. **Identify the given information**:
- \( a \parallel b \)
- \( m \angle 1 = 80^\circ \)
2. **Understand the relationships**:
- Vertically opposite angles are equal. Therefore, \( m \angle 1 = m \angle 4 \) and \( m \angle 5 = m \angle 8 \).
3. **Recognize parallel lines with a transversal**:
- Alternate interior angles are equal. Therefore, \( m \angle 1 = m \angle 6 \) and \( m \angle 4 = m \angle 5 \).
- Corresponding angles are equal. Therefore, \( m \angle 1 = m \angle 5 \) and \( m \angle 4 = m \angle 8 \).
4. **Calculating other angles**:
- Since \( m \angle 1 = 80^\circ \), by corresponding angles, \( m \angle 5 = 80^\circ \).
- Since \( m \angle 5 = 80^\circ \) and vertically opposite angles are equal, \( m \angle 8 = 80^\circ \).
#### Conclusion
Therefore, \( m \angle 8 = 80^\circ \).
When a transversal intersects two parallel lines, it creates equal corresponding and alternate interior angles. This understanding helps to determine the measure of unknown angles based on known measures.
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