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.

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 angle B and angle D 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:

  • means one thing is congruent to another
  • means an angle
  • 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:

[insert drawing of ∠DOG and ∠CAT, identically constructed using rays and having equal angles of 55° but oriented in different directions]

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

  • O  A
  • O = A = 55°

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

If you need the measure in radians, you will write 0.959931 rad.

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 DOG and 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?:

[insert drawings of ∠HEN made with line segments, ∠PIG made with lines, ∠FOX made with rays, and ∠ANT made with a line segment and a ray; all should be pointing in different rotations]

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:

[insert drawing of ∠YAK using either line segments YA and AK or rays AY and AK]

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 Point M:
  2. [insert drawing of horizontal Ray MU, with Point M as the origin]

  3. 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.
  4. [insert drawing or photograph of this step]

  5. Without changing the compass, place the point of the compass on Point M on your new drawing. Swing an arc from Point M up into the space above your new ray.
  6. [insert drawing or photograph of this step]

  7. 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 Point K and reach Point Y with it.
  8. [insert drawing or photograph of this step]

  9. Without changing the compass, move the compass point to the new ray's point, here Point U, and swing the arc that intersects with your original arc.
  10. [insert drawing or photograph of this step]

  11. Use your straightedge to connect the vertex, here Point M, with the intersection of the two arcs. You have copied the existing angle.
  12. [insert drawing or photograph of this step]

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.

[insert drawing of protractor]

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:

[insert drawing of two 45° angles oriented in 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 1.8 rad. Is the angle congruent to anything?
  5. Look at the drawings. Can you find two congruent angles?

[insert drawing of five angles, two of which are congruent but all five angles are oriented (turned) in different directions; label the two congruent ones GNU and OWL; the other three could be SOW, RAT, and ELK]

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 1.8 rad is congruent to itself.
  5. The two congruent angles are GNU and OWL.

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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