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.
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
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.