Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.
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 = ∠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.
From Opposite Angles of a Parallelogram are Equal to HOME PAGE
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.
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.