Question

In a parallelogram any two consecutive angles are supplementary

Answer 1

Step-by-step explanation:

To prove : ∠A   + ∠B = 180°

First, we suppose that ABCD is a parallelogram. Compare  

ΔABC   and  ΔCDA

1. AC = AC (common side)

2.  ∠CAB  =  ∠ACD   ...    (alternate interior angles)

3.  ∠DAC =  ∠BCA   ...    (alternate interior angles)

Thus, the two triangles are congruent, which means that  

∠B   =  ∠D

Similarly, we can show that  ∠A   =  ∠C.

This proves that opposite angles in any parallelogram are equal.

Sum of all angles of a quadrilateral is 360°.

∠A   + ∠B + ∠C +  ∠D = 360°

⇒ ∠A   + ∠B + ∠A   + ∠B  = 360°

⇒ 2 ∠A   + 2 ∠B  = 360°

⇒ 2 (∠A   + ∠B ) = 360°

⇒ ∠A   + ∠B  = 360 / 2

⇒ ∠A   + ∠B = 180°

Similarly, we can prove that,

∠B + ∠C= 180°   ;   ∠A   + ∠D = 180°    ;   ∠D + ∠C= 180°

Hence Proved