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 = ∠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
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.
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