What does it mean when we say to sides or angles are congruent?

What Are Congruent Angles?

Congruent angles are two or more angles that are identical to one another (and to themselves). Congruent angles can exist acute, obtuse, exterior, or interior angles. Information technology does non thing what type of angle you accept; if the measure of angle one is the aforementioned as angle ii, they are congruent angles.

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

Table of Contents

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1. Congruent Angles Definition
• Coinciding Angles Symbol
• Reflexive Holding of Congruence
2. Congruent Angles Examples
3. Drawing Coinciding Angles
4. Coinciding Angles Discussion Problems

Congruent Angles Definition

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

If $a\mathrm{due\; north}\mathrm{grand}l\mathrm{due\; east}B$ and $an\mathrm{one\; thousand}leD$ have the same mensurate, they are said to have congruency.

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

Coinciding Angles Symbol

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

• $\cong$ means 1 affair is congruent to another
• $\angle$ means an bending
• $\measuredangle$ is sometimes used to bespeak a measured bending
• $°$, every bit in $45°$, means degrees
• $rad$ means radians, a method of measuring angles in the metric system

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

We could say that $\angle O$ ($angleO$) and $\angle A$ ($an\mathrm{one\; thousand}leA$) are congruent, and both measure $55°$. We could also say that mathematically:

• $\angle O\cong \angle A$
• $\angle O=\angle A=55°$

Since both angles measure less than $\mathrm{ninety}°$, they are also acute and are both made using rays. The autograph description, $\angle O$ and $\angle A$ identifies each angle'south vertex, or point where rays come across.

If you lot demand the measure in radians, you lot will write $0.959931rad$.

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, and so they have specific shapes and sizes. Therefore every bending is congruent to itself.

Congruent Angles Examples

Angles tin exist oriented in whatsoever management on a plane and withal be coinciding. Just as $\angle DOG$ and $\angle CAT$, in a higher place, were congruent but were not "lined upwardly" with each other, then besides can congruent angles announced 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 management — the manner the ii angles sit on the printed page or screen — is unimportant. The manner the two angles are constructed is unimportant. If the measures in degrees or radians are equal, the angles are congruent.

Cartoon Congruent Angles

Y'all tin can draw coinciding angles, or compare possible existing congruent angles, using a cartoon compass, a straightedge, and a pencil.

One of the easiest means to draw congruent angles is to draw ii parallel lines cut past a transversal. In your drawing, the respective angles will be congruent. You lot volition have multiple pairs of angles with congruency.

Some other easy way to draw congruent angles is to depict a right angle or a right triangle. And then, cutting that correct angle with an angle bisector. If yous bisect the angle exactly, you are left to ii congruent acute angles, each measuring $45°$.

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

Just what if you have a given angle and need to draw an identical (coinciding) bending next to it:

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

Here are the steps for how to draw coinciding angles:

1. Draw a ray to the right of your original angle, just some distance away. Create an endpoint for your ray and label it. We will call ours $Poi\mathrm{due\; north}tM$.
2. Open your drawing compass so that the point on the compass can be placed on the vertex of the existing angle, merely the pencil does non reach past the drawn line segments or rays of the existing bending.
3. Without changing the compass, place the point of the compass on $Poi\mathrm{due\; north}t\mathrm{Thou}$ on your new drawing. Swing an arc from $PointM$ upward into the space above your new ray.
4. Move the compass point to a point on ane ray of the original angle, then adjust the cartoon compass, so the pencil touches the other point. Here we put our compass on $Poi\mathrm{northward}tK$ and reach $Poi\mathrm{due\; north}tY$ with information technology.
5. Without irresolute the compass, move the compass signal to the new ray's signal, here $PointU$, and swing the arc that intersects with your original arc.
6. Apply your straightedge to connect the vertex, here $PointM$, with the intersection of the ii arcs. You accept copied the existing bending.

If you demand to compare 2 angles that are not labeled with their degrees or radians, you lot tin similarly employ a drawing compass to locate points on both angles and measure their degree of openness.

If you do not accept a protractor handy, you lot tin use institute objects to get a sense of an angle's measurement. The square edge of a sail of paper is $90°$. If you fold that corner over so the 2 sides exactly line up, you accept 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 unlike directions.

Coinciding Angles Word Problems

1. Suppose yous are told that 2 angles of two different triangles are congruent. Does that hateful the triangles must be congruent?
2. 1 angle measures $91°$ and is synthetic of 2 rays. Another angle measures $91°$ merely 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 coinciding?
4. An bending measures $\mathrm{i.eight}rad$. Is the angle coinciding to annihilation?

Just as whatever bending is true to itself by being congruent, be true to yourself by doing the work kickoff, before checking out the answers!

1. Two angles of ii different triangles can be coinciding, simply that does non mean you lot accept congruent triangles; they could be different sizes, and their other angles could take dissimilar measures.
2. The two angles, ane measuring $91°$ and constructed of two rays and the other, also measuring $91°$ merely constructed of two line segments, are congruent. Just the bending matters.
3. Two angles, each measuring $47°$, are congruent, no affair how they are synthetic.
4. An bending measuring $\mathrm{1.viii}rad$ is congruent to itself.

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Source: https://tutors.com/lesson/congruent-angles

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