Question

if a quadrilateral each pair of opposite angles is equal then it is a parallelogram. prove step by step​

Answer 1

EXPLANATION.

→ ABCD is a quadrilateral in Which opposite

sides are parallel.

→ < A = < C and < B = < D

→ To prove → AB || CD and AD || BC.

→ sum of all angles of a quadrilateral is

equal to 360°.

→ < A + < B + < C + < D = 360°.

→ < A + < D + < A + < D = 360°.

[ < A = < C and < B = < D = GIVEN ]

→ 2 < A + 2 < D = 360°.

→ < A + < D = 180°.

→ AB || DC.

→ Similarly,

→ < A + < B + < C + < D = 360°.

→ < A + < B + < A + < B = 360°.

[ < A = < C and < B = < D = GIVEN ]

→ 2 < A + 2 < B = 360°.

→ < A + < B = 180°.

→ AD || BC

→ AB || DC AND AD || BC.

→ So, ABCD is a parallelogram.

Answer 2

Question

if a quadrilateral each pair of opposite angles is equal then it is parallelogram. Prove step by step.

Given

• ABCD is a quadrilateral whose opposite angles are equal
• example - ∠A = ∠C and ∠B = ∠D

Required 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

So,

= ∠A + ∠B + ∠C + ∠D = 360°

= ∠A ∠B + ∠A + ∠B = 360°

= ( ∠A = ∠C and ∠B = ∠D )

= 2 ∠A + 2 ∠B = 369°

= ∠A + ∠B = 180°

AD || BC

therefore AB || DC and AD || BC

Hence ABCD is parallelogram.

More

Properties of Parallelogram

• Opposite sides are congruent ( AB = DC )
• Each diagonal of a parallelogram separates it into two congruent triangles.
• The diagonal of parallelogram bisect each other.
• Consecutive angles are supplemtary ( A + D = 180° )
• If one angle is right, then all the angles are right
• Opposite angles are congruent ( D = B )