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.
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.
The easiest way to measure the number of degrees in an angle is with a protractor.
To talk and write about or draw angles, we need common symbols and words to describe them. We have three symbols mathematicians use:
Let's look at how we can describe these two angles:
We could say that () and () are congruent, and both measure . We could also say that mathematically:
Since both angles measure less than , they are also acute and are both made using rays. The shorthand description, and identifies each angle's vertex, or point where rays meet.
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.
Angles can be oriented in any direction on a plane and still be congruent. Just as and , 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.
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 .
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:
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 . If you fold that corner over so the two sides exactly line up, you have a 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.
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!
After working your way through this lesson and video, you have learned:
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