Question

if in a quadrilateral each pair of opposite angles are equal then it is a parallelogram .....prove it without using angle sum property

Answer 1

Answer:

Correct question is : If opposite angles of a quadrilateral is equal then it is a parallelogram.

Given that ABCD is a quadrilateral whose opposite angles are equal

i.e. ∠A =∠C and ∠B = ∠D

To prove: ABCD is a parallelogram

Proof:

As we know that sum of angle of Quadrilateral is 360°

⇔∠A+∠B∠C+D = 360°

∵ ∠A =∠C and ∠B = ∠D , ⇔∠A+∠D+∠A+∠D =360°

⇔2∠A +2∠D = 360°

⇒2∠A = 2∠D =360°

⇔∠A+∠D = 180° ( Co-interior angle)

⇔AB║DC

Similarly

∠A+∠B+∠C+∠D = 360°

⇔∠A+∠B+∠A+∠B = 360° (∠A = ∠C and ∠B = ∠D)

⇔2∠A+2∠B = 360°

⇔∠A+∠B = 180°

⇔ AD║BC

therefore AB║DC and AD║BC

hence ABCD is a parallelogram

Answer 2

Answer:

Given:- Quadrilateral ABCD

angle A = angle C; angle B = angle D... [1]

To prove:- The given figure is a parallelogram

angle A + angle B + angle C + angle D = 360*

angle A + angle D + angle A + angle D = 360* (from [1])

angle A + angle D = 180*... [2]

AB || DC (from [2] as angle A and angle D become co-interior angles)

Similarly, we can prove that AD || BC

Hence proved - The given quadrilateral ABCD is a parallelogram by definition.

hope it helps...

plzz mark me as the brainliest!!