All the students of 9th grade must know the properties of a parallelogram before proving the theorem on Opposite Angles of a Parallelogram is Equal. If you understand the theorem then you can solve different types of problems on properties of a parallelogram. We know that the opposite sides of a parallelogram are the same and parallel. Draw a transversal line from A to C in a parallelogram ABCD. The alternate angles of a parallelogram become equal and it is divided into two triangles.

## Opposite Angles of a Parallelogram are Equal Theorem

**Question:** Prove that opposite angles of a parallelogram are equal?

Given: In a Parallelogram, the Opposite Angles Are Equal.

Proof:

Given: ABCD is a parallelogram, and ∠A, ∠B, ∠C, ∠D are the four angles.

To Prove that:

∠A =∠C and ∠B=∠D

Let us assume that ABCD is a parallelogram. Now compare triangles ABC, and CDA. Here we have

AC=AC (common sides)

∠1=∠4 (alternate interior angles)

∠2=∠3 (alternate interior angles).

Thus, by ASA, the two triangles are congruent, which means that ∠B=∠D. Similarly, we can show that ∠A=∠C. This proves that opposite angles in any parallelogram are equal.

**Example:**

If one angle of a parallelogram is 90°, show that all its angles will be equal to 90°.

**Solution:**

Consider the parallelogram ABCD in which ∠A is a right angle

We know that in any parallelogram, the opposite angles are equal.

Therefore ∠C=90°.

Also, in any parallelogram, the adjacent angles are supplementary.

This implies ∠B=180° – ∠ A=180° – 90° =90°.

Similarly

∠D=180° – ∠C=180° – 90° =90°.

Hence, ∠A=∠B=∠C=∠D = 90°.

All the angles in this parallelogram are equal to 90°.

We have proved that when one angle of a parallelogram is 90°, the parallelogram is a rectangle.

Also, See:

- Pair of Opposite Sides of a Parallelogram are Equal and Parallel
- Opposite Sides of a Parallelogram are Equal

**Points to Remember:**

1. Opposite sides of a parallelogram are congruent.

2. Opposite angles of a parallelogram are congruent.

3. Diagonals of a parallelogram bisect each other.

4. The consecutive angles are supplementary.

5. In a parallelogram if any of the angles is a right angle, then all the angles will be right angles.

### FAQs on Opposite Angles of a Parallelogram are Equal

**1. How do you find the opposite angles of a parallelogram?**

There are 5 properties to be known to find the opposite angles of a parallelogram

1. Opposite sides are congruent

2. Opposite angles are congruent

3. Diagonals bisect each other.

4. Consecutive angles are supplementary

5. If any of the angles is a right angle, then all the angles will be right angles.

**2. Are all angles of parallelogram equal?**

A parallelogram must have equivalent opposite interior angles. The sum of all interior angles must have equal degrees. As the angles and are opposite interior angles they must be equivalent.

**3. Is opposite angles are equal?**

Opposite angles are non-adjacent angles formed by two intersecting lines. Opposite angles are congruent.