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

Proof:

            Statement:

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)

            Reason


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

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