Opposite Angles of a Parallelogram are Equal

Here we will discuss about the opposite angles of a parallelogram are equal.

In a parallelogram, each pair of opposite angles are equal.

Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS

To prove: ∠P = ∠R and ∠Q = ∠S

Construction: Join PR and QS.

Opposite Angles of a Parallelogram are Equal



In ∆PQR and ∆RSP;

1. ∠QPR = ∠PRS

2. ∠QRP = ∠SPR

3. ∠QPR + ∠SPR = ∠PRS + ∠QRP

⟹ ∠P = ∠R

4. Similarly, from ∆PQS and ∆RSQ, ∠Q = ∠S.     (Proved)


1. PQ ∥ SR and PR is a transversal.

2. QR ∥ PS and PR is a transversal.

3. Adding statements 1 and 2.

Converse proposition of the above theorem

A quadrilateral is a parallelogram if each pair of opposite angles are equal.

Given: PQRS is a quadrilateral in which ∠P = ∠R and ∠Q = ∠S

Pair of Opposite Angles are Equal

To prove: PQRS is a parallelogram

Proof: ∠P + ∠Q + ∠R + ∠S = 360°, because the sum of the four angles of a quadrilateral is 360°.

Therefore, ∠2P + ∠2Q = 360°, (since ∠P = ∠R, ∠Q = ∠S)

Therefore, ∠P + ∠Q = 180° and so, ∠P + ∠S = 180°, (since ∠Q = ∠S)

∠P + ∠Q = 180°

⟹ PS ∥ QR (since sum of the co. interior angles is 180°)

∠P + ∠S = 180°

⟹ PQ ∥ SR (since sum of the co. interior angles is 180°)

Therefore, in the quadrilateral PQRS, PQ ∥ SR and PS ∥ QR. So, PQRS is a parallelogram.

9th Grade Math

From Opposite Angles of a Parallelogram are Equal to HOME PAGE

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?

Recent Articles

  1. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 06, 23 01:23 AM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More

  2. Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

    Dec 04, 23 02:14 PM

    Different types of Indian Coins
    Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

    Read More

  3. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Dec 04, 23 01:50 PM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More