# Angles of a Parallelogram

A quadrilateral whose two pairs of sides are parallel to each and the four angles at the vertices are not equal to the right angle, then the quadrilateral is called a parallelogram. Also, the opposite sides are equal in length.

Here,

AB = CD (opposite sides)

Sum of all the four angles = 360 degrees

The important properties of angles of a parallelogram are:

• If one angle of a parallelogram is a right angle, then all the angles are right angles
• Opposite angles of a parallelogram are equal (or congruent)
• Consecutive angles are supplementary angles to each other (that means they add up to 180 degrees)

## Opposite Angles of a Parallelogram

In the above parallelogram, A, C and B, D are a pair of opposite angles.

## Opposite Angles of a Parallelogram are equal

Theorem: Prove that the opposite angles of a parallelogram are equal.

Given: Parallelogram ABCD.

To prove: ∠B = ∠D and ∠A=∠C

Proof:

In the parallelogram ABCD,

AB \\ CD and AD \\ BC

Consider triangle ABC and triangle ADC,

AC = AC (common side)

We know that alternate interior angles are equal.

∠1 = ∠4

∠2 = ∠3

By ASA congruence criterion, two triangles are congruent to each other.

Therefore, ∠B = ∠D and ∠A=∠C

Hence, it is proved that the opposite angles of a parallelogram are equal.

## Consecutive Angles of a Parallelogram

Theorem: Prove that any consecutive angles of a parallelogram are supplementary.

Given: Parallelogram ABCD.

To prove: ∠A + ∠B = 180 degrees, ∠C + ∠D = 180 degrees

Proof:

AB ∥ CD and AD is a transversal.

We know that interior angles on the same side of a transversal are supplementary.

Therefore, ∠A + ∠D = 180°

Similarly, ∠B + ∠C = 180°, ∠C + ∠D = 180° and ∠A + ∠B = 180°.

Therefore, the sum of any two adjacent angles of a parallelogram is equal to 180°.

Hence, it is proved that any two adjacent or consecutive angles of a parallelogram are supplementary.

If one angle is a right angle, then all four angles are right angles:

From the above theorem, it can be decided that if one angle of a parallelogram is a right angle (that is equal to 90 degrees), then all four angles are right angles. Hence, it will become a rectangle.

Since, the adjacent sides are supplementary.

For example, ∠A, ∠B are adjacent angles and ∠A = 90°, then:

∠A + ∠B = 180°

90° + ∠B = 180°

∠B = 180° – 90°

∠B = 90°

Similarly, ∠C = ∠D = 90°

Example 1:

In the adjoining figure, ∠D = 85° and ∠B = (x+25)°, find the value of x.

Solution:

Given,

∠D = 85° and ∠B = (x+25)°

We know that, opposite angles of a parallelogram are congruent or equal.

Therefore,

(x+25)° = 85°

x = 85° -25°

x = 60°

Hence, the value of x is 60.

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