What are Angles?
Two rays or line segments that meet at a point is called an angle. The two lines of an angle are called the arms of the angle and the point at which the rays or lines meet is called the vertex. Angles are measured in terms of degrees or radians. To measure angles in degrees, we use a protractor that is either a plastic transparent semi-circle with markings on it, or a full circle with markings on it.
What are Congruent Angles?
As mentioned earlier, congruence is the property demonstrated by equal shapes, lines and angles. On the same lines, two angles are congruent when they measure the same.
In these images there are two angles, ∠ABC and ∠DEF. It is seen that even though the angles are constructed differently, the measure of the two angles is the same, which is 136°. This indicates that these two angles are congruent.
Congruent Angles with the Help of Parallel Lines
We can also use parallel lines with a transversal drawn across them to observe congruent angles. In the image we can see eight angles that are formed when a transversal EF intersects two parallel lines AB and CD.
Let’s identify the congruent angles in the image:
- There are four pairs of corresponding angles that are congruent:
- ∠2 and ∠4
- ∠8 and ∠6
- ∠5 and ∠7
- ∠1 and ∠3
- There are four pairs of vertical angles that are congruent:
- ∠3 and ∠6
- ∠4 and ∠5
- ∠2 and ∠7
- ∠1 and ∠8
So, with the help of parallel lines and a transversal intersecting the parallel lines we observe that there are eight pairs of congruent angles.
Rapid Recall
Solved Examples
Example 1:
Identify the congruent angles in the following pairs of angles:
Solution:
Both the angles measure 39°.
Hence, they are congruent angles.
2. 
The two angles have different measures, 42° and 39°.
Hence, they are not congruent angles.
3. 
The two angles have different measures, 106° and 103°.
Hence, they are not congruent angles.
4. 
The two angles have different measures, 19° and 21°.
Hence, they are not congruent angles.
When we compare two regular polygons, that is shapes made of straight lines and with sides of equal lengths, they can be congruent if they fit perfectly when placed upon one another.
Lines are one dimensional, meaning that there is only length without the dimensions of height and width. Two lines are congruent only when their lengths are the same.
Similarity is related to equal angles, but the shapes being compared may not have an equal length when referring to the sides of the shapes being compared. On the other hand, when referring to the congruence of shapes, it indicates that the shape has equal angles as well as sides that are equal in length.