Given: line a || line b. If m3 2x + 10 and mZ5 5x 40, then m25 %3D |3D 3 4 5 6 7 8 30* O 110" O 150* 70°

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
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### Parallel Lines and Angle Measures

**Given:**  
- Line \( a \parallel \) Line \( b \).

**Problem Statement:**  
*If \( m\angle 3 = 2x + 10 \) and \( m\angle 5 = 5x - 40 \), then what is \( m\angle 5 \)?*

**Diagram Explanation:**
There is a diagram showing two parallel lines, \( a \) and \( b \), with a transversal crossing them. The angles formed at the intersections are labeled as follows:

```
     1   2
      \ /
       a
     3   4
------\------
       /
      b
     5   6
      / \
     7   8
```
Angle pairs are formed between the two parallel lines and the transversal:

- Angles \(1\) and \(5\) are corresponding angles.
- Angles \(3\) and \(5\) are alternate interior angles.
- Other pairs of angles follow similar relationships due to the parallel nature of lines \(a\) and \(b\).

**Solution Approach:**

1. **Identify Corresponding Angles:**
   Since angles 3 and 5 are alternate interior angles, they are equal when lines \(a\) and \(b\) are parallel. Therefore, we can set up the equation:
   \[
   2x + 10 = 5x - 40
   \]

2. **Solve for \( x \):**
   \[
   2x + 10 = 5x - 40
   \]
   Subtract \( 2x \) from both sides:
   \[
   10 = 3x - 40
   \]
   Add 40 to both sides:
   \[
   50 = 3x
   \]
   Divide by 3:
   \[
   x = \frac{50}{3}
   \]
   \[
   x \approx 16.67
   \]

3. **Calculate \( m\angle 5 \):**
   Substitute \( x \) back into the expression for \( m\angle 5 \):
   \[
   m\angle 5 = 5x - 40
   \]
   \[
   m\angle 5 = 5\left(\frac{50}{3
Transcribed Image Text:### Parallel Lines and Angle Measures **Given:** - Line \( a \parallel \) Line \( b \). **Problem Statement:** *If \( m\angle 3 = 2x + 10 \) and \( m\angle 5 = 5x - 40 \), then what is \( m\angle 5 \)?* **Diagram Explanation:** There is a diagram showing two parallel lines, \( a \) and \( b \), with a transversal crossing them. The angles formed at the intersections are labeled as follows: ``` 1 2 \ / a 3 4 ------\------ / b 5 6 / \ 7 8 ``` Angle pairs are formed between the two parallel lines and the transversal: - Angles \(1\) and \(5\) are corresponding angles. - Angles \(3\) and \(5\) are alternate interior angles. - Other pairs of angles follow similar relationships due to the parallel nature of lines \(a\) and \(b\). **Solution Approach:** 1. **Identify Corresponding Angles:** Since angles 3 and 5 are alternate interior angles, they are equal when lines \(a\) and \(b\) are parallel. Therefore, we can set up the equation: \[ 2x + 10 = 5x - 40 \] 2. **Solve for \( x \):** \[ 2x + 10 = 5x - 40 \] Subtract \( 2x \) from both sides: \[ 10 = 3x - 40 \] Add 40 to both sides: \[ 50 = 3x \] Divide by 3: \[ x = \frac{50}{3} \] \[ x \approx 16.67 \] 3. **Calculate \( m\angle 5 \):** Substitute \( x \) back into the expression for \( m\angle 5 \): \[ m\angle 5 = 5x - 40 \] \[ m\angle 5 = 5\left(\frac{50}{3
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