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Show that opposite angles of parallelogram are equal. 

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Updated on : 2022-09-05

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Here, ABCD is a parallelogram with AC as its diagonal.

We know, in parallelogram opposites sides are parallel.

So, AB∥DC and AD∥BC

Since, AB∥DC and AC is the transversal

⇒  ∠BAC=∠DCA    ---- ( 1 )  [ Alternate angles ]

Similarly, AD∥BC and AC is the transversal.

⇒  ∠DAC=∠BCA     ---- ( 2 )  [ Alternate angles ]

Adding ( 1 ) and ( 2 ),

⇒  ∠BAC+∠DAC=∠DCA+∠BCA

⇒  ∠BAD=∠DCB

Similarly, we can prove, ∠ADC=∠ABC

∴  We have proved that, opposite angles of a parallelogram are equal.

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Angles of a Parallelogram

Know about angles of a parallelogram, representation, properties, related theorems and solved examples online. Find the remaining angles when one angle has been given.

MathsMath ArticleAngles Of A Parallelogram

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, and then the quadrilateral is called a parallelogram. Also, the opposite sides are equal in length.

Here,

AD = BC (opposite sides)

AB = CD (opposite sides)

Sum of all the four angles = 360 degrees

Learn more about the parallelogram here.

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)

Read more:

Area of Parallelogram

Lines And Angles Class 7

Parallel Lines Transversals Angle

Perimeter of Parallelogram

Opposite Angles of a Parallelogram

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

Therefore, ∠A = ∠C and ∠B = ∠D

Also, we have different theorems based on the angles of a parallelogram. They are explained below along with proofs.

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 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°

Solved Examples

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.

Example 2: Observe the below figure.

Find the values of x, y and z.

Solution:

From the given figure,

y = 112° {since the opposite angles of a parallelogram are equal)

z + 40° + 112° = 180° {the sum of consecutive angles is equal to 180°}

z = 180° – 112° – 40° = 28°

Also, x = 28° {z and x are alternate angles)

Therefore, x = 28°, y = 112° and z = 28°.

Example 3: 

Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find their measures.

Solution: Given,

The ratio of two adjacent angles of a parallelogram  = 4 : 5

Let 4x and 5x the angles.

Then, 4x + 5x = 180° {the sum of two adjacent angles of a parallelogram is supplementary}

9x = 180° x = 20°

Therefore, the angles are 4 × 20° = 80° and 5 × 20° = 100°.

Practice Problems

Two adjacent angles of a parallelogram are in the ratio 5 : 1. Find all the angles of the parallelogram.

The opposite angles of a parallelogram are (3x – 4)° and (2x – 1)°. Find the measures of all angles of the parallelogram.

If one of the interior angles of a parallelogram is 100°, then find the measure of all the remaining angles.

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Prove that: In a parallelogram, opposite angles are equal.

Prove that: In a parallelogram, opposite angles are equal.. Ans: Hint: We know that the opposite sides of the parallelogram is parallel. Now drawing a transversal line AC, in the parallelogram ABCD, the alternate angle becomes equal and we get two di...

Prove that: In a parallelogram, opposite angles are equal.

Last updated date: 15th Mar 2023

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Hint: We know that the opposite sides of the parallelogram is parallel. Now drawing a transversal line AC, in the parallelogram ABCD, the alternate angle becomes equal and we get two different equations. Adding these both equations, at last , we prove that the opposite angles of the parallelogram are equal.Complete step-by-step answer:

It is already given in the question that a parallelogram ABCD has AC as its one of the diagonal.

To prove:- ∠ ∠ A = ∠ ∠ C and ∠ ∠ B = ∠ ∠ D.

Proof:- Opposite sides of parallelogram is parallel.

So, AB ∥ ∥ CD and AD ∥ ∥ BC. Since, AB ∥ ∥ CD;

And AC is the transversal.

∠ ∠ BAC = ∠ ∠

DCA ( They are the Alternate angles)

So, ∠ ∠ BAC = ∠ ∠ DCA …(1) Now, Since, AD ∥ ∥ BC;

And AC is the transversal.

∠ ∠ DAC = ∠ ∠

BCA (They are the Alternate angles)

So, ∠ ∠ DAC = ∠ ∠ BCA ….(2)

Adding both the equations, that is equation (1) and (2) , we get;

⇒ ⇒ ∠ ∠ BAC + ∠ ∠ DAC = ∠ ∠ DCA + ∠ ∠ BCA ⇒ ⇒ ∠ ∠ BAD = ∠ ∠ DCB. ⇒ ⇒ ∠ ∠ A = ∠ ∠ C.

In the similar way only;

We can prove that:- ∠ ∠ ADC = ∠ ∠ ABC ⇒ ⇒ ∠ ∠ D = ∠ ∠ B.

Hence, it is proved that in a parallelogram , the opposite sides of parallelogram are equal.Note: In order to solve this particular question, we need to memorize these properties of parallelogram:-

a.) The opposite sides are congruent.

b.) The opposite angles are congruent.

c.) The consecutive angles are supplementary.

d.) If anyone of the angles is a right angle, then all the other angles will be the right angle.

e.) The two diagonals bisect each other.

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