Last modified on April 22nd, 2021

chapter outline

 

Congruent Angles

What are Congruent Angles

Two or more angles that have exactly the same measure are called congruent angles. In other words they have the same number of degrees. Congruent angles can be an acute, obtuse, exterior, or interior angles.

In geometry ‘congruent’ means identical or same to one another in shape and size. Here, it is important to remember that the direction of the angle or the length of their edges has no effect on congruency.

Congruent Angles

Its Symbol

The symbol for congruence is ≅

In the above figure, the pair of congruent angles is represented as

∠ABC ≅ ∠XYX

How to Find Congruent Angles

The best way to find whether two or more angles are congruent is to measure the given angles for congruency using a protractor.

Congruent Angle Examples

Angles can be oriented in any direction on a plane keeping them congruent to each other. Shown below are congruent angles all measuring 48° but each having a different orientation.

Thus, ∠ABC ≅ ∠EFG ≅ ∠IJK ≅ ∠LMN

Show with 48° in each one them.

∠ABC ≅ ∠EFG ≅ ∠IJK ≅ ∠LMN

Congruent Angles Examples

Constructing Congruent Angles

To draw congruent angles we need a compass, a straight edge, and a pencil.

One of the easiest ways to draw congruent angles is to make a transversal that cuts two parallel lines. The multiple pairs of corresponding angles formed are congruent.

Another common way of drawing congruent angles is to draw a right angle or right-angled triangle. Bisecting the right angle with an angle bisector we will get two congruent acute angles each measuring 45°.

But, when we need to draw an angle that is congruent to a given angle we need to follow the steps given below:

Let us assume that we are drawing a congruent angle to ∠YAK.

  1. Draw a ray to the right of the given angle. Mark an endpoint to the ray and label it as A.
  2. The point on the compass is then placed on the vertex of the existing angle but the pencil should not reach past the line drawn or rays of the existing angle.
  3. Without changing the position of the compass place the point of the compass on point A and then draw an arc from point A to the space above the new ray.
  4. Place the compass point over a point on one ray of the original angle and then adjust the compass such that the pencil touches the other point. Here place the compass point over point K and reach point Y with it.
  5. Without changing the position of the compass points, move the compass to the ray point X and swing an arc that intersects with the original arc.
  6. Use the straightedge to connect the vertex, here point A, with the intersection of the two arcs. The new congruent angle is formed.

We can measure the two angles for comparison using a compass.

Ways of Proving Angles Congruent Using Appropriate Theorems

There are different theorems to prove whether two or more angles are congruent. The theorems are given below:

1. Congruent Complements Theorem: It states that if two angles are complements of the same angle, then the two angles are congruent.

2. Congruent Supplements Theorem: It states that if two angles are supplements of the same angle, then the two angles are congruent.

3. Alternate Interior Angles Theorem: It states that if a transversal intersects two parallel lines, then the alternate interior angles are congruent.

4. Converse of the Alternate Interior Angles Theorem: It states that if the alternate interior angles formed when a transversal intersects two lines are congruent, then the lines are parallel.

5. Alternate Exterior Angles Theorem: It states that if a transversal intersects two parallel lines, then the alternate exterior angles formed are congruent.

6. Converse of the Alternate Exterior Angles Theorem: It states that if the alternate exterior angles formed when a transversal intersects two lines are congruent, then the lines are parallel.

7. Corresponding Angles Theorem: It states that if a transversal intersects two parallel lines, then the corresponding angles formed are congruent.

8. Converse of the Corresponding Angles Theorem: It states that if the corresponding angles formed when a transversal intersects two lines are congruent, then the lines are parallel.

9. Overlapping Angle Theorem: It states that if two angles adjacent to a common angle are congruent, the overlapping angles formed are congruent.

Two angles are congruent. The measure of the first angel is (2x + 4) °, and the second angle is 94°. Find the value of x.

Solution:

As we know,
If two angles are congruent, the measure of their angles is equal.
Thus,
(2x + 4)° = 94°
2x = 94°- 4°
x = 90°/2
x = 45°

In a triangle two of the three angles are congruent and the third angle is 40°. What are the measures of the congruent angles.

Solution:

As we know,
The sum of the angles of a triangle = 180°
Let the measure of the two congruent angles = x
Thus in the given triangle, we can write,
40° + x + x = 180°
2x = 180° – 40°
2x = 140°
x =140°/2 = 70°
Thus the measure of the two congruent angles in the given triangle is 70°

Last modified on April 22nd, 2021

Leave a Reply

Your email address will not be published. Required fields are marked *