Math Problem Statement

Given: m∠1 = 2x + 40, m∠2 = 2y + 40, m∠3 = x + 2y. Find m∠1, m∠2, and m∠3.

Solution

Let's solve the problem step by step.

We are given:

  • m1=2x+40m\angle 1 = 2x + 40
  • m2=2y+40m\angle 2 = 2y + 40
  • m3=x+2ym\angle 3 = x + 2y

From the diagram, angles 1\angle 1 and 2\angle 2 appear to be vertically opposite angles, which means they are congruent (equal). Therefore, we can set their measures equal to each other: 2x+40=2y+402x + 40 = 2y + 40 By subtracting 40 from both sides: 2x=2y2x = 2y Dividing both sides by 2: x=yx = y

Now that we know x=yx = y, we can substitute this into the equation for m3m\angle 3: m3=x+2y=y+2y=3ym\angle 3 = x + 2y = y + 2y = 3y

Next, we know that the sum of the angles on a straight line is 180°. Thus: m1+m3=180m\angle 1 + m\angle 3 = 180^\circ Substituting m1=2x+40m\angle 1 = 2x + 40 and m3=3ym\angle 3 = 3y, and knowing x=yx = y: (2y+40)+3y=180(2y + 40) + 3y = 180 Simplifying the equation: 5y+40=1805y + 40 = 180 Subtracting 40 from both sides: 5y=1405y = 140 Dividing by 5: y=28y = 28

Now that we have y=28y = 28, and x=yx = y, so x=28x = 28, we can find the measures of the angles:

  • m1=2x+40=2(28)+40=56+40=96m\angle 1 = 2x + 40 = 2(28) + 40 = 56 + 40 = 96^\circ
  • m2=2y+40=2(28)+40=96m\angle 2 = 2y + 40 = 2(28) + 40 = 96^\circ
  • m3=3y=3(28)=84m\angle 3 = 3y = 3(28) = 84^\circ

Thus, the measures of the angles are:

  • m1=96m\angle 1 = 96^\circ
  • m2=96m\angle 2 = 96^\circ
  • m3=84m\angle 3 = 84^\circ

Would you like more details on any step or further explanations?

Here are five related questions to explore:

  1. How do you identify vertically opposite angles in any geometric figure?
  2. Can you solve a problem where angles are given in terms of different variables?
  3. How does the concept of supplementary angles help in solving linear pairs of angles?
  4. What are other ways to verify that 1\angle 1 and 2\angle 2 are equal in the diagram?
  5. What happens if we change the expressions for the angles? How does the solution process adjust?

Tip: In geometry, identifying angle relationships such as vertical angles, corresponding angles, and supplementary angles is key to solving many problems involving intersecting lines.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Angle Relationships

Formulas

m∠1 = 2x + 40
m∠2 = 2y + 40
m∠3 = x + 2y

Theorems

Vertical Angles Theorem
Linear Pair Theorem
Supplementary Angles Theorem

Suitable Grade Level

Grades 9-11