Geometry

GED Mathematical Reasoning: Angles

Labeling an angle

In some situations, an angle is labeled with three letters, one on each ray and one at the vertex. In this case, we refer to the angle using three letters, where the second letter is the vertex. For example, here is an angle called “angle ABC” and it measures 32 degrees.

In other situations, an angle is labeled with one letter located on the inside of the angle. Here is an angle called angle “a” and it measures 25 degrees.

Complementary angles

Notice that, together, angles $\angle BAC$ and $\angle CAD$ form the 90 degree angle. We actually have a special name for angles that add to be 90 degrees. They are called COMPLIMENTARY angles.

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Supplementary angles

Angles that add to be 180 degrees are called SUPPLEMENTARY angles. So we would say that $\angle FEG$ and $\angle GEH$ are SUPPLEMENTARY angles.

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Vertical angles

In order to complete example three it is important to know the definition and property of VERTICAL angles. Vertical angles are opposite of each other and have the same angle measure. For instance, angles d and b are opposite each other and therefore are called vertical angles.

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Example 1

Using the diagram, state the measure of: $\angle CAD$

There are three angles in the diagram.

$\angle BAC$ , which we’re told measures 35 degrees.

$\angle CAD$ , which is the angle measure we’re being asked to find.

And $\angle BAD$ , which measures 90 degrees and so is a right angle, as indicated by the small square at the vertex.

Notice that, together, angles $\angle BAC$ and $\angle CAD$ form the 90 degree angle.

$\angle BAC + \angle CAD = 90^{\circ}$

$\angle BAC$ and $\angle CAD$ are complimentary angles and their sum is 90 degrees. We know that $\angle BAC$ measures 35 degrees, therefore, the measure of $\angle CAD$ equals the difference between 90 degrees and 35 degrees.

90 degrees minus 35 degrees equals 55 degrees, making the measure of $\angle CAD$ 55 degrees.

$\angle CAD = 90^{\circ} -35^{\circ} = 55^{\circ}$

Example 2

Using the diagram on the right, state the measure of: $\angle FEG$

Notice that there are three angles shown in the diagram for Example Two.

$\angle FEG$ , the angle whose measure we’re being asked to find.

$\angle GEH$ , which we’re told measures 51 degrees.

And $\angle FEH$ , which is a straight angle, measuring 180 degrees.

Together, angles $\angle FEG$ and $\angle GEH$ form the 180 degree angle.

$\angle FEG + \angle GEH = 180^{\circ}$

$\angle FEG$ and $\angle GEH$ are supplementary angles and their sum is 180 degrees. We know that $\angle GEH$ measures 51 degrees, therefore, the measure of $\angle FEG$ equals the difference between 180 degrees and 51 degrees.

180 degrees minus 51 degrees equals 129 degrees, which means $\angle FEG$ measures 129 degrees.

$\angle FEG = 180^{\circ} - 51^{\circ} = 129^{\circ}$

Example 3

Using the diagram on the right, state the measure of $\angle c$

The diagram in example three contains many angles, but only four angles are labeled. They are labeled as lowercase a, b, c, and d.

Angles a and c are also vertical angles, because they are opposite each other. Since vertical angles have the same measure and we know that angle a measures 123 degrees, then that means angle c measures 123 degrees as well.

Before we conclude Example 3, let’s talk about one more definition related to angles. Referring back to the diagram, angles a and b are called ADJACENT angles because they share a side.

There are three more pairs of adjacent angles in this diagram, can you identify them?

They are: angles b and c… angles c and d … and angles d and a.

Notice that these pairs of adjacent angles combine to form a straight angle with a measure of 180 degrees. So not only are they adjacent, they are supplementary angles as well.

$\angle a + \angle b = 180^{\circ}$
$\angle c + \angle b = 180^{\circ}$
$\angle c + \angle d = 180^{\circ}$
$\angle a + \angle a = 180^{\circ}$

From here, we can determine the measure of the remaining two angles in the diagram – angles d and b.

Since adjacent angles a and b are supplementary, their sum is 180 degrees. We know the angle a measures 123 degrees. Therefore, angle b equals the difference between 180 degrees and 123 degrees. 180 degrees minus 123 degrees equals 57 degrees, therefore angle b measures 57 degrees.

$\angle b = 180^{\circ} - 123^{\circ} = 57^{\circ}$

And since angles b and d are vertical angles, and vertical angles are equal, then angle d also measures 57 degrees.