Math Problem Statement

answer it

What is the measure of ∠3 given m∠1 = x - 6 and m∠2 = 5x + 6?

Solution

The problem involves two intersecting lines, and we are given:

m1=x6andm2=5x+6\text{m} \angle 1 = x - 6 \quad \text{and} \quad \text{m} \angle 2 = 5x + 6

To solve this, follow these steps:

  1. Vertical Angle Theorem: Angles 1 and 3 are vertical angles, so they are equal. This means:

    m1=m3=x6\text{m} \angle 1 = \text{m} \angle 3 = x - 6

  2. Linear Pair: Angles 1 and 2 form a linear pair, meaning their measures add up to 180°. Therefore:

    m1+m2=180\text{m} \angle 1 + \text{m} \angle 2 = 180^\circ

    Substituting the expressions for the angles:

    (x6)+(5x+6)=180(x - 6) + (5x + 6) = 180

  3. Simplify the equation:

    x6+5x+6=180x - 6 + 5x + 6 = 180

    Combine like terms:

    6x=1806x = 180

  4. Solve for xx:

    x=1806=30x = \frac{180}{6} = 30

  5. Find the measure of 3\angle 3: Since m3=x6\text{m} \angle 3 = x - 6:

    m3=306=24\text{m} \angle 3 = 30 - 6 = 24^\circ

So, the measure of 3\angle 3 is 2424^\circ.

If you need any clarification, feel free to ask! Here are some related questions:

  1. What is the value of 2\angle 2?
  2. Can you explain the Vertical Angle Theorem in detail?
  3. How does the Linear Pair Postulate work in this context?
  4. What if the given angles were supplementary instead of vertical?
  5. Can we use this method to solve other angle relationships?

Tip: Always look for angle relationships (vertical, linear pair, etc.) when dealing with intersecting lines!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Angle Relationships

Formulas

m∠1 + m∠2 = 180°

Theorems

Vertical Angle Theorem
Linear Pair Postulate

Suitable Grade Level

Grades 8-10