## What Are Congruent Angles?

Congruent angles are two or more angles that are identical to one another (and to themselves). Congruent angles can be acute, obtuse, exterior, or interior angles. It does not matter what type of angle you have; if the measure of angle one is the same as angle two, they are congruent angles.

Congruent in geometry means that one figure, whether it is (line segment, polygon, angle, or 3D shape), is identical to another in shape and size. Corresponding angles on congruent figures are always congruent.

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## Congruent Angles Definition

The definition of congruent angles is two or more angles with equal measures in degrees or radians. Congruent angles need not face the same way or be constructed using the same figures (rays, lines, or line segments). If the two angle measurements are equal, the angles are congruent.

If and have the same measure, they are said to have congruency.

The easiest way to measure the number of degrees in an angle is with a protractor.

### Congruent Angles Symbol

To talk and write about or draw angles, we need common symbols and words to describe them. We have three symbols mathematicians use:

• $\cong$ means one thing is congruent to another
• $\angle$ means an angle
• $\measuredangle$ is sometimes used to indicate a measured angle
• $°$, as in $45°$, means degrees
• $rad$ means radians, a method of measuring angles in the metric system

Let's look at how we can describe these two angles:

We could say that $\angle O$ () and $\angle A$ () are congruent, and both measure $55°$. We could also say that mathematically:

Since both angles measure less than $90°$, they are also acute and are both made using rays. The shorthand description, $\angle O$ and $\angle A$ identifies each angle's vertex, or point where rays meet.

If you need the measure in radians, you will write .

### Reflexive Property of Congruence

The Reflexive Property of Congruence tells us that any geometric figure is congruent to itself. A line segment, angle, polygon, circle, or another figure of the given size and shape is self-congruent.

Angles have a measurable degree of openness, so they have specific shapes and sizes. Therefore every angle is congruent to itself.

## Congruent Angles Examples

Angles can be oriented in any direction on a plane and still be congruent. Just as $\angle DOG$ and $\angle CAT$, above, were congruent but were not “lined up” with each other, so too can congruent angles appear in any way on a page.

Here is a drawing that has several angles. Which of these angles are congruent?:

All of these angles are congruent.

The direction — the way the two angles sit on the printed page or screen — is unimportant. The way the two angles are constructed is unimportant. If the measures in degrees or radians are equal, the angles are congruent.

## Drawing Congruent Angles

You can draw congruent angles, or compare possible existing congruent angles, using a drawing compass, a straightedge, and a pencil.

One of the easiest ways to draw congruent angles is to draw two parallel lines cut by a transversal. In your drawing, the corresponding angles will be congruent. You will have multiple pairs of angles with congruency.

Another easy way to draw congruent angles is to draw a right angle or a right triangle. Then, cut that right angle with an angle bisector. If you bisect the angle exactly, you are left to two congruent acute angles, each measuring $45°$.

Two $45°$ angles are congruent complementary angles. Complementary angles are congruent only if the angles measure $45°$.

But what if you have a given angle and need to draw an identical (congruent) angle next to it:

Here are the steps for how to draw congruent angles:

1. Draw a ray to the right of your original angle, but some distance away. Create an endpoint for your ray and label it. We will call ours .
2. Open your drawing compass so that the point on the compass can be placed on the vertex of the existing angle, but the pencil does not reach past the drawn line segments or rays of the existing angle.
3. Without changing the compass, place the point of the compass on on your new drawing. Swing an arc from up into the space above your new ray.
4. Move the compass point to a point on one ray of the original angle, then adjust the drawing compass, so the pencil touches the other point. Here we put our compass on and reach with it.
5. Without changing the compass, move the compass point to the new ray's point, here , and swing the arc that intersects with your original arc.
6. Use your straightedge to connect the vertex, here , with the intersection of the two arcs. You have copied the existing angle.

If you need to compare two angles that are not labeled with their degrees or radians, you can similarly use a drawing compass to locate points on both angles and measure their degree of openness.

If you do not have a protractor handy, you can use found objects to get a sense of an angle's measurement. The square edge of a sheet of paper is $90°$. If you fold that corner over so the two sides exactly line up, you have a $45°$ angle.

The position or orientation of two angles has nothing to do with their congruence. Angles can be congruent while facing in two different directions.

## Congruent Angles Word Problems

1. Suppose you are told that two angles of two different triangles are congruent. Does that mean the triangles must be congruent?
2. One angle measures $91°$ and is constructed of two rays. Another angle measures $91°$ but is constructed of two line segments. Are the two angles congruent?
3. Two angles are each $47°$, but one is made from a line and ray, the other is made from a line segment and a line. Are the two angles congruent?
4. An angle measures . Is the angle congruent to anything?

Just as any angle is true to itself by being congruent, be true to yourself by doing the work first, before checking out the answers!

1. Two angles of two different triangles can be congruent, but that does not mean you have congruent triangles; they could be different sizes, and their other angles could have different measures.
2. The two angles, one measuring $91°$ and constructed of two rays and the other, also measuring $91°$ but constructed of two line segments, are congruent. Only the angle matters.
3. Two angles, each measuring $47°$, are congruent, no matter how they are constructed.
4. An angle measuring is congruent to itself.

## What you learned:

After working your way through this lesson and video, you have learned:

• Congruent Angles Definition
• How to indicate congruent angles using symbols
• Find congruent angles
• The reflexive property of congruence
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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