Lesson 13

Using Equations to Solve for Unknown Angles

Let’s figure out missing angles using equations.

13.1: Is This Enough?

Tyler thinks that this figure has enough information to figure out the values of \(a\) and \(b\).

Two rays on the same side of line l meet at the same point to form 3 angles, a, 90 degrees, b.

Do you agree? Explain your reasoning.

13.2: What Does It Look Like?

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: \(x = 35\)

    Diego: \(x+35=180\)

    Two lines meet to form 4 angles, 35 degrees, x degrees, w degrees, blank.
  2. Elena: \(35+w+41=180\) 

    Diego: \(w+35=180\)

    Three adjacent angles form a straight angle, the angles are labeled 35 degrees, w degrees, 41 degrees.
  3. Elena: \(w + 35 = 90\)

    Diego: \(2w+35=90\)

    Two angles, w degrees and 35 degrees, appear to be complementary. Another angle, w degrees, is adjacent to the 35 degree angle.
  4. Elena: \(2w + 35 = 90\)

    Diego: \(w+35=90\)

    A right angle is split into three angles, w degrees, blank, w degrees. A 35 degree angle is formed by two rays outside the right angle and is vertical to the blank angle.
  5. Elena: \(w + 148 = 180\)

    Diego: \(x+90=148\)

    A set of rays form the angles, clockwise, 148 degrees, w, x, blank, blank, blank. 148 and w are supplementary, w and x are complementary, the next two blanks sum to 90 degrees.

13.3: Calculate the Measure

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

Two lines meet to form 4 angles.  One set of adjacent angles is labeled w degrees, 124 degrees.
Two rays on the same side of line l meet at point Q to form 3 angles, 52 degrees, b degrees, 23 degrees.


Lines \(\ell\) and \(m\) are perpendicular.

Two lines and a ray meet at the point where line m is perpendicular to line l.  Ask for additional assistance.
Two lines form vertical angles, one is labeled 120 degrees, the other is split by rays into three angles labeled m degrees, 66 degrees, m degrees.


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 \(a+b+c\).

The diagram contains 3 squares. Segments connect the bottom left corner of the diagram to the top right corners the squares.  Please ask for additional assistance.
  1. Use a protractor to measure the three angles. Use your measurements to conjecture about the value of \(a+b+c\).
  2. Find the exact value of \(a+b+c\) by reasoning about the diagram.


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 \(x\) in this diagram.

Two lines meet to form 4 angles. An angle is labeled 144 degrees. It's vertical angle is split into 4 smaller angles, x degrees, x degrees, x degrees, 90 degrees.

Using what we know about vertical angles, we can write the equation \(3x + 90 = 144\) to represent this situation. Then we can solve the equation.

\(\begin{align} 3x + 90 &= 144 \\ 3x + 90 - 90 &= 144 - 90 \\ 3x &= 54 \\ 3x \boldcdot \frac13 &= 54 \boldcdot \frac13 \\ x &= 18 \end{align}\)