Lesson 13Using Equations to Solve for Unknown Angles

Learning Goal

Let’s figure out missing angles using equations.

Learning Targets

  • I can write an equation to represent a relationship between angle measures and solve the equation to find unknown angle measures.

Lesson Terms

  • adjacent angles
  • complementary
  • right angle
  • straight angle
  • supplementary
  • vertical angles

Warm Up: Is This Enough?

Problem 1

Tyler thinks that this figure has enough information to figure out the values of and .

Straight line l with 2 intersecting lines. The three angles as a, 90, and b.

Do you agree? Explain your reasoning.

Activity 1: What Does It Look Like?

Problem 1

Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles.

  1. Elena:

    Diego: 

    Two lines intersecting to make a vertical x. The top angle is 35 degrees and the bottom is w. Angle x is on the right.

  2. Elena:  

    Diego: 

    A straight line with two lines intersecting. The three angles are 35, w, and 41.

  3. Elena: 

    Diego:

    Four lines intersection making angles w, 35, and w.

  4. Elena:

    Diego:

    Three lines intersecting making angles w, unmarked, w, unmarked, 35, and unmarked.

  5. Elena:
    Diego:

    Two lines intersecting to make a vertical x. The top angle is 35 degrees and the bottom is w. Angle x is on the right.

Activity 2: Calculate the Measure

Problem 1

Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.

  1. Two lines intersecting making an "x" shape. Angles marked are w and adjacent to w is 124.
  2. Three lines intersecting making three angles: 52, b, and 23.

  3. Lines and are perpendicular.

    Perpendicular lines l and m intersected. by 3 other lines forming angles x, 23, 4 unmarked angles, x, and 2 more unmarked angles.
  4. Four lines intersecting forming angles 120, unmarked, m, 66, m, unmarked

Are you ready for more?

Problem 1

The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of .

A diagram of three squares. Three additional segments have been drawn that connect corners of the squares - a, b, and c.

  1. Use a protractor to measure the three angles. Use your measurements to conjecture about the value of .

  2. Find the exact value of  by reasoning about the diagram.

Lesson Summary

To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of in this diagram.

A horizontal line with lined intersecting it at a center point. Angles formed are unmarked, x, x, x, 90, unmarked, 144.

Using what we know about vertical angles, we can write the equation to represent this situation. Then we can solve the equation.