# 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.

Proof:

 Statement:In ∆PQR and ∆RSP;1. ∠QPR = ∠PRS2. ∠QRP = ∠SPR3. ∠QPR + ∠SPR = ∠PRS + ∠QRP⟹ ∠P = ∠R 4. Similarly, from ∆PQS and ∆RSQ, ∠Q = ∠S.     (Proved) Reason1. 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

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.

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.

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

Dec 06, 23 01:23 AM

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

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

Dec 04, 23 02:14 PM

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: