https://file.aiquickdraw.com/mathbot/file/1729871144306tkc4lood.jpg

The image shows two intersecting lines with angles formed. The problem is asking to find the value of \( x \) and the measure of angle \( \angle 1 \) (denoted as \( m \angle 1 \)).

### Step-by-Step Solution:

We have the following given expressions for the angles:

- \( 3x + 5 \) for one of the angles
- \( 5x - 39 \) for the angle marked as \( \angle 1 \)

Since these two angles are vertical angles, they are congruent. Therefore, we can set their expressions equal to each other:

\[
3x + 5 = 5x - 39
\]

### Solving for \( x \):

1. Move the terms involving \( x \) to one side:
\[
3x - 5x = -39 - 5
\]

2. Simplify:
\[
-2x = -44
\]

3. Solve for \( x \):
\[
x = \frac{-44}{-2} = 22
\]

Thus, \( x = 22 \).

### Finding the measure of \( \angle 1 \):

Substitute \( x = 22 \) into either angle expression (since both are equal, you can use either one).

Using the expression for \( \angle 1 \):

\[
m \angle 1 = 5x - 39 = 5(22) - 39 = 110 - 39 = 71^\circ
\]

Thus, \( m \angle 1 = 71^\circ \).

### Final Answers:
- \( x = 22 \)
- \( m \angle 1 = 71^\circ \)

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to explore:
1. What are vertical angles, and why are they always congruent?
2. How do you know when angles are supplementary or congruent?
3. What other properties can you infer from intersecting lines and their angles?
4. Can you generalize the steps to solve for \( x \) when two angles are given as algebraic expressions?
5. How would the solution change if the lines were parallel and we were dealing with corresponding angles?

**Tip:** Always check for angle relationships, such as vertical angles or supplementary angles, to set up equations when dealing with intersecting lines.

On a separate sheet of paper, find the value of x and m∠1.

Learn how to find the value of x and the measure of angle 1 using vertical angles theorem. This problem involves solving linear equations and applying geometric concepts. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation