Vertical angles and how to find their measures

When two straight lines cross, they make four angles around one point. The angles across from each other have the same measure. Let’s use that relationship to find a missing value, then use it to fill in an angle inside a triangle.

Find the opposite pairs at an intersection

Vertical angles are the opposite angles made by two intersecting lines. They share a vertex, and the two angles in each pair are congruent, which means their measures are equal.

Angles a and b are one vertical pair. Angles c and d are another.

Angles $a$ and $c$ form a straight angle, so their measures add to $180^{°}$ . Angles $b$ and $c$ also form a straight angle. Subtract the same measure, $m ∠ c$ , from $180^{°}$ in both cases and you get equal measures for $a$ and $b$ . The same reasoning works for the other pair.

Set two vertical angle measures equal

The labels are on opposite sides of the intersection.

The labeled angles are vertical, so $(10 x + 18)^{°}$ and $58^{°}$ are equal. Solve for the number $x$ .

\begin{array}{l} 10 x + 18 = 58 \\ 10 x = 40 \\ x = 4 \end{array}

With $x = 4$ , each angle measures $58^{°}$ .

Use a vertical angle inside a right triangle

The angle marked outside the triangle has the same measure as the angle at the left corner inside.

The angle labeled $(4 x + 8)^{°}$ is across the intersection from the angle inside the triangle. So the triangle has angle measures $(4 x + 8)^{°}$ , $(3 x + 12)^{°}$ , and $90^{°}$ . Their sum is $180^{°}$ .

\begin{array}{l} (4 x + 8) + (3 x + 12) + 90 \\ = 180 \\ 7 x + 110 = 180 \\ 7 x = 70 \Rightarrow x = 10 \end{array}

At $x = 10$ , the two acute angles measure $48^{°}$ and $42^{°}$ . Together with the right angle, they add to $180^{°}$ .

Learn mathKrista KingMay 18, 2019math, learn online, online course, online math, geometry, congruent angles, vertical angles, intersecting lines, angles at the intersection point

Vertical angles and how to find their measures

Find the opposite pairs at an intersection

Set two vertical angle measures equal

Use a vertical angle inside a right triangle

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