Congruence of Angles: Congruent angles are the angles that have equal measure. So all the angles that have the same measure will be known as congruent angles. Congruent angles are seen everywhere, for instance, in isosceles triangles, equilateral triangles, or when a transversal crosses two parallel lines. Let us learn more about the congruent angles along with different theorems based on congruent angles.

What is Congruence of Angles?

In mathematics, the definition of congruent angles is “angles that are equal in the measure are known as congruent angles”. To put it differently, equal angles are congruent angles. It is denoted by the symbol \(≅\), so if we want to represent \(∠A\) is congruent to \(∠B\), we will write it as \(∠A ≅ ∠B\). Look at a congruent angles example given below.

In the above image, both the angles are equal in measurement (\(35°\) each). They can completely overlap each other. So, we can say that both the given angles are congruent as per the definition.

Examples of Congruence of Angles

While \(∠G\) and \(∠S\) are not facing the same direction, we can see that they have the same measure of \(38°\), and hence, they are congruent.
While \(∠R\) and \(∠Q\) have edges with different lengths, we can see that they have the same measure of \(152°\), and hence, they are congruent.
Observing the following four figures, we notice that in the first figure, the arms of the angle is made of two line segments \(EH\) and \(EN\). In the second figure, the angle is made of two lines \(PI\) and \(IG\). In the third figure, the angle is made of two rays \(OF\) and \(OX\) while in the fourth figure, one arm is a line segment \(AN\) and another arm is a ray \(AT\).

Irrespective of what the angle is made of, \(∠HEN ≅ ∠PIG ≅ ∠FOX ≅ ∠TAN\), as the measurement of each angle is \(63°\).
Here are some examples of congruent angles in different shapes

In the above figure, each regular polygon has all the interior angles congruent.
The measure of each angle in a regular triangle or equilateral triangle is \(60°\).
In a square measure of each angle is \(90°\).
In a pentagon measure of each angle is \(108°\).
While in a regular hexagon, the measure of an interior angle is \(120°\) and so on.
These examples remind us that regardless of the length of the angles, edges or the direction the angles are facing, as long as the angles have the same measure, they are considered congruent.

Theorems Based on Congruence of Angles

There are many theorems based on congruent angles. Applying the congruent angles theorem, we can find out easily whether two angles are congruent or not. Those theorems are listed below:

1. Vertical angles theorem
2. Corresponding angles theorem
3. Alternate angles theorem
4. Congruent supplements theorem
5. Congruent complements theorem

Vertical Angles Theorem

Statement: Vertical angles (the angles formed when two lines intersect each other) are congruent.
Vertical angles are always congruent, as stated by the vertical angles theorem.
Previously we know that: Angles on a straight-line sum up to \(180°.\)

\(∠1 + ∠4 = 180°\) ……(1) (Linear Pair)
\(∠1 + ∠2 = 180°\) ……(2) (Linear Pair)
\(∴ ∠1 + ∠4 = ∠1 + ∠2\) [By equating LHS of equation (1) and (2)]
\(∴ ∠1 + ∠4 = ∠1 + ∠2\) (Quantities equal to the same quantity are equal to each other).
(Transitive: if \(x = y\) and \(y = z\) that implies \(x = z\))
\(∴ ∠4 = ∠2\) (By eliminating \(∠1\) on both sides)
Also, \(∠1 = ∠3\) (Similarly, we can prove for \(∠1\) and \(∠3\))
Thus, \(∠4 ≅ ∠2\) and \(∠1 ≅ ∠3\)
Conclusion: Vertically opposite angles are always congruent.

Corresponding Angles Theorem

By the definition of the corresponding angles, when two parallel lines are cut by a third one, the angles that occupy the same relative position at each intersection are known to be corresponding angles to each other.

When a transversal \(p\) intersects two parallel lines \(1\) and \(m\), corresponding angles are always congruent to each other. In this figure, \(∠1 ≅ ∠2\). As it’s a postulate, we need not prove this. Without proof, it is always stated as true.

Alternate Interior Angles Theorem

When a transversal \(p\) intersects two parallel lines \(l\) and \(m\), each pair of alternate interior angles are congruent.
Referring to the above figure.
We have:
\(∠1 = ∠5\) ……(1) (corresponding angles)
\(∠3 = ∠5\) ……(2) (vertically opposite angles)
Thus, \(∠1 = ∠3\) [Equating LHS of equation (1) and (2)]

Similarly, we can prove the other pairs of alternate interior angles is congruent too.
Other pair of angles from the figure, \(∠2 = ∠4\)
Thus, \(∠1 ≅ ∠3\) and \(∠2 ≅ ∠4\) are the pair of alternate interior angles.

Congruent Supplements Theorem

Supplementary angles are those whose sum is \(180°\) According to this theorem, angles supplementing to the same angle are congruent, whether they are adjacent angles or not.

We can prove this theorem by applying the linear pair property of angles, as,
\(∠1 + ∠2 = 180°\) …….(1) (Linear pair of angles)
\(∠2 + ∠3 = 180°\) …….(2) (Linear pair of angles)
From the above equation (1) and (2), we get, \(∠1 = ∠3\)
\(∴\) Angles supplement to the same angle are congruent angles.

Congruent Complements Theorem

Complementary angles are those whose sum is \(90°.\) According to this theorem, angles complementary to the same angle are congruent angles, whether they are adjacent angles. Let us understand it with the help of the following figure below.

We can prove this theorem easily as both the angles formed are right angles.
\(∠a + ∠b = 90°\) ……(1) (\(∵ ∠a\) and \(∠b\) form \(90°\) angle)
\(∠a + ∠c = 90°\) ……(2) (\(∵ ∠a\) and \(∠c\) form \(90°\) angle)
So, from the above equation (1) and (2), we get, \(∠b ≅ ∠c\)
\(∴\) Two angles that are complementary to the same angle are congruent angles.

How Do You Solve Congruent Angles?

Let us consider two given angles \((3x + 54)°\) and \((10x + 40)°\) are congruent
If two angles are congruent, the measure of their angles is the same.
\(⇒ (3x + 54)° = (10x + 40)°\)
\(⇒ 10x° – 3x° = 54° – 40°\)
Solving for \(x\), we have that \(7x° = 14°\) and so \(x° = 2°\).
To find the measure of the angles, substitute \(x° = 2°\) back into the expressions for the angle measures.
We have, \(3x° + 54° = 3 × 2° + 54° = 60°\) and \(10x° + 40° = 10 × 2° + 40° = 60°\)
Thus, the measure of each congruent angle is \(60°\).

Solved Examples on Congruence of Angles

Q.1. \(∠ABC\) and \(∠PQR\) are congruent and \(∠ABC = 55°\), then what will be the measurement of \(∠PQR\)?
Ans: Given, \(∠ABC ≅ ∠PQR\) and \(∠ABC = 55°\)
The measure of congruent angles is equal
\(∴ ∠PQR = 55°\)
Thus, the measurement of \(∠PQR = 55°\).

Q.2. If \(∠ROT ≅ ∠TOS ≅ ∠SOP\), then find the angle which is congruent to \(∠SOR\).

Ans: Given, \(∠ROT ≅ ∠TOS ≅ ∠SOP\)
Let us consider, \(∠ROT ≅ ∠SOP\)
Adding \(∠TOS\) on both sides, we get,
\(⇒ ∠ROT + ∠TOS ≅ ∠SOP + ∠TOS\)
\(⇒ ∠SOR ≅ ∠TOP\)
Thus, the angle congruent to \(∠SOR\) is \(∠TOP\).

Q.3. State the condition of congruency for two angles.
Ans: For two or more angles to be congruent, their angle measure should be equal.

Q.4. Write down the properties of congruence of angles.
Ans: Properties:
1. Every single angle is congruent to itself-that is \(∠A ≅ ∠A\).
2. If \(∠X\) and \(∠Y\) are two angles and \(∠X ≅ ∠Y\), then \(∠Y ≅ ∠X\).
3. If \(∠P,\,∠Q\) and \(∠R\) are three angles and \(∠P ≅ ∠Q\) and \(∠Q ≅ ∠R\), then \(∠R ≅ ∠P\).

Q.5. Is it correct to say that any two right angles are congruent? Give a reason to justify your answer.
Ans: Right angles are always congruent as their measurement is the same. They always measure \(90°.\)

Q.6. How many sides can two congruent angles share?
Ans: Two congruent angles can share one, two, or no sides.

Q.7. Are adjacent angles congruent?
Ans: Adjacent angles are congruent only when their common side bisects their sum. That is, the common side must be an angle bisector.

Summary

In this article, we learnt that for a pair of angles to be congruent, the measure of both angles should be equal. We discussed some of the examples where the angles are congruent such as equilateral triangles and regular polygons like pentagon hexagon etc. Then moving further, we learned the proof of congruence of angles that are vertical angles theorem, corresponding angles theorem, alternate angles theorem, congruent supplements theorem, and congruent complements theorem.

And we learnt how the angles are congruent based on these theorems. We studied when two unknown angles are said to be congruent and how to find the measure of angles. Lastly, we solved some different types of examples of the concept of congruence of angles.

FAQs on Congruence of Angles

Q.1. What are congruent angles?
Ans: Congruent angles are the angles that have equal measure. So all the angles that have the same measure will be congruent angles.

Q.2. What are the different theorems based on the congruence of angles?
Ans: There are many theorems based on congruent angles. Those theorems are listed below:
1. Vertical angles theorem
2. Corresponding angles theorem
3. Alternate angles theorem
4. Congruent supplements theorem
5. Congruent complements theorem

Q.3. How do you know if angles are congruent?
Ans: Two angles are congruent if they have the same measure irrespective of the arms they are made of and irrespective of the length of their arms.

Q.4. Do congruent angles add up to \(180°\)?
Ans: In general, all congruent angles are not supplementary angles. For angles to sum up to \(180°\), they must be supplementary angles. So right angles are congruent as well as supplementary angles because they have the same measure and they sum up to \(180°.\)

Q.5. What shape has congruent angles?
Ans: In all the regular polygons such as equilateral triangle, square, pentagon, hexagon,  the angles are equal in measure and, thus, congruent to each other. Also, the isosceles triangle has base angles congruent, a rectangle has all the angles right angles, and all different types of parallelogram such as rhombus, kite have opposite angles congruent.

Q.6. What shapes have all angles congruent?
Ans: In all regular polygons such as equilateral triangle, square, pentagon, hexagon, the angles are equal in measure and, thus, congruent to each other. Also, a rectangle has all angles congruent.
The following figures show shapes with all angles congruent: