Lesson 5

Using Equations to Solve for Unknown Angles

Let’s figure out missing angles using equations.

5.1: 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.

5.2: 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.

5.3: 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.

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 \(x\) 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.

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

Glossary Entries

  • adjacent angles

    Adjacent angles share a side and a vertex.

    In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

    Three segments all joined at endpoint B. Point A is to the left of B and segment A B is drawn. Point C is above B and segment C B is drawn. Point D is to the right of B and segment B D is drawn.
  • complementary

    Complementary angles have measures that add up to 90 degrees.

    For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.

    complementary angles of 15 and 75 degrees
    Two angles, one is 75 degrees and one is 15 degrees
  • right angle

    A right angle is half of a straight angle. It measures 90 degrees.

    a right angle
  • straight angle

    A straight angle is an angle that forms a straight line. It measures 180 degrees.

    a 180 degree angle
  • supplementary

    Supplementary angles have measures that add up to 180 degrees.

    For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.

    supplementary angles of 15 and 165 degrees
    supplementary angles of 15 and 165 degrees
  • vertical angles

    Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

    For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

    Angles \(AED\) and \(BEC\) are another pair of vertical angles.

    a pair of intersecting lines that create vertical angles