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Lesson 5

Using Equations to Solve for Unknown Angles

Let’s figure out missing angles using equations.

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.

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Do you agree? Explain your reasoning.

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.

Elena: $x = 35$

Diego: $x+35=180$
Elena: $35+w+41=180$

Diego: $w+35=180$
Elena: $w + 35 = 90$

Diego: $2w+35=90$
Elena: $2w + 35 = 90$

Diego: $w+35=90$
Elena: $w + 148 = 180$

Diego: $x+90=148$

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.

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Two rays on the same side of line l meet at point Q to form 3 angles, 52 degrees, b degrees, 23 degrees.

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Two lines and a ray meet at the point where line m is perpendicular to line l. Ask for additional assistance.

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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.

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Are you ready for more?

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.

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Use a protractor to measure the three angles. Use your measurements to conjecture about the value of $a+b+c$ .
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.

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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.

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

Video VLS G7U7V1 Angle Relationships (Lessons 1–5) available at https://player.vimeo.com/video/516923320.

adjacent angles

Adjacent angles share a side and a vertex.

In this diagram, angle $ABC$ is adjacent to angle $DBC$ .
complementary

Complementary angles have measures that add up to 90 degrees.

For example, a $15^\circ$ angle and a $75^\circ$ angle are complementary.
right angle

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

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

Supplementary angles have measures that add up to 180 degrees.

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