Math Problem Statement

solve

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

Solution

The image shows two intersecting lines with angles formed. The problem is asking to find the value of xx and the measure of angle 1\angle 1 (denoted as m1m \angle 1).

Step-by-Step Solution:

We have the following given expressions for the angles:

  • 3x+53x + 5 for one of the angles
  • 5x395x - 39 for the angle marked as 1\angle 1

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

3x+5=5x393x + 5 = 5x - 39

Solving for xx:

  1. Move the terms involving xx to one side: 3x5x=3953x - 5x = -39 - 5

  2. Simplify: 2x=44-2x = -44

  3. Solve for xx: x=442=22x = \frac{-44}{-2} = 22

Thus, x=22x = 22.

Finding the measure of 1\angle 1:

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

Using the expression for 1\angle 1:

m1=5x39=5(22)39=11039=71m \angle 1 = 5x - 39 = 5(22) - 39 = 110 - 39 = 71^\circ

Thus, m1=71m \angle 1 = 71^\circ.

Final Answers:

  • x=22x = 22
  • m1=71m \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 xx 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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Linear Equations

Formulas

3x + 5 = 5x - 39

Theorems

Vertical Angles Theorem

Suitable Grade Level

Grades 8-10