Math Problem Statement

∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, 4, x, minus, 4, right parenthesis, degrees∠1=(4x−4) ∘ and mangle, 2, equals, left parenthesis, 5, x, minus, 29, right parenthesis, degrees∠2=(5x−29) ∘ , then find the measure of angle, 1∠1.

Solution

Given that ∠1 and ∠2 are vertical angles, their measures are equal. Therefore, we can set up the equation:

m1=m2m∠1 = m∠2

This means:

4x4=5x294x - 4 = 5x - 29

Step-by-Step Solution:

  1. Set up the equation: 4x4=5x294x - 4 = 5x - 29

  2. Subtract 4x4x from both sides: 4=x29-4 = x - 29

  3. Add 29 to both sides: 25=x25 = x

  4. Now, substitute x=25x = 25 back into the expression for m1m∠1: m1=4x4=4(25)4=1004=96m∠1 = 4x - 4 = 4(25) - 4 = 100 - 4 = 96^\circ

Thus, the measure of ∠1 is:

m1=96m∠1 = 96^\circ


Would you like further details or have any questions?

Related Questions:

  1. How can you prove that vertical angles are always equal?
  2. What are supplementary angles, and how do they relate to vertical angles?
  3. What if ∠1 was expressed as 3x23x - 2 instead—how would that change the solution?
  4. How does this approach change when solving for more complex angle relationships?
  5. How do parallel lines influence the relationship between angles?

Tip: Always check if angles are supplementary or complementary when solving problems with intersecting lines!

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Vertical Angles

Formulas

Equation setup: m∠1 = m∠2
Solving linear equations: ax + b = cx + d

Theorems

Vertical Angles Theorem: Vertical angles are equal

Suitable Grade Level

Grades 8-10