Question

abcd is a trapezium in which ab is parallel to cd and ad is eaual to bc prove that angle a is equal to angle b and angle c is equal to angle d​

Answer 1

Answer:

Step-by-step explanation:

Given: ABCD is a trapezium, in which AB || DC and AD = BC.

To Prove: (i) ∠A = ∠B  

(ii) ∠C = ∠D  

(iii) △ABC ≅ △BAD  

(iv) Diagonal AC = diagonal BD.

Const.: Produce AB to E, such that a line through C parallel to DA (CE || DA), intersects AB in E.

Proof : (i) In quad. AECD

AE || DC [given]

and AD || EC [by const.]

⇒ AECD is a Parallelogram  

⇒ AD = EC [opp side ||gm]

But, AD = BC [given]

⇒ BC = EC

⇒ ∠2 = ∠1 [angle opp. to equal sides of a △]

Also, ∠1 + ∠3 = 180° [linear pair]

and ∠A + ∠2 = 180° [consecutive interior ∠s]

⇒ ∠A + ∠2 = ∠1 + ∠3  

⇒ ∠A = ∠3  [∵ ∠2 = ∠1]

or ∠A = ∠B

(ii) AB || DC and AD is a transversal  

∴ ∠A + ∠D = 180° [consecutive interior angle]  

⇒ ∠B + ∠D = 180° ---(i)[∵∠A = ∠B]

Again, AB || DC and BC is transversal .

∴ ∠B + ∠C = 180°  

From (i) and (ii), we have  

∠B + ∠C = ∠B + ∠D  

⇒ ∠C = ∠D