Question

ABCD is a trapezium in which AB is parallel to CD and AD is equal to BC. show that angle A is equal to angle B

Answer 1

Given :- ABCD is a trapezium

AB || CD

AD = BC

To proof :-

(i)∠A = ∠B

(ii)∠C = ∠D

(iii)∆ ABC ≅ ∆ BAD

(iv)Diagonal AC = Diagonal = BD

Construction :- Draw DA || CE

Solution :-

(i) Since it's given ABCD is a trapezium

AB || CD

DA || CE ( By construction)

Therefore, ADCE is a parallelogram

So, DA = CE &

DC = AE ( Opposite side of parallelogram are equal )

But, AD = BC

Therefore, BC = CE ( Given )

∠CEB = ∠CBE ( In ∆ CBE angles opposite to equal sides are equal )

180° - ∠DAB = 180° - ∠ABC

[ ADCE is a parallelogram and ∠A + ∠E = 180° ∠B & ∠CBE form a linear pair ]

∠A = ∠B ( Cancelling 180° from both sides)

(ii) Co interior angles on the same side of a transversal are supplementary

∠A + ∠D = 180° & ∠B + ∠C = 180°

∠A + ∠D = ∠B + ∠C

∠B + ∠D = ∠B + ∠C ( ∠A = ∠B proved above)

∠D = ∠C

(iii) In ∆ ABC & ∆ BAD

AB = BA

∠B = ∠A ( proved above )

BC = BD ( Given )

∆ ABC ≅ ∆ BAD ( By SAS criteria)

(iv) AC = BD ( CPCT )

Answer 2

Let us extend AB. Then, draw a line through C, which is parallel to AD, intersecting AE at point E

AD||CE ( by construction)

AE||DC ( as AB is extended to E)

It is clear that AECD is a parallelogram

(i) AD = CE (Opposite sides of parallelogram AECD)

However,

AD = BC (Given)

Therefore,

BC = CE

∠CEB = ∠CBE (Angle opposite to equal sides are also equal)

Consider parallel lines AD and CE. AE is the transversal line for them

∠A + ∠CEB = 1800 (Angles on the same side of transversal)

∠A + ∠CBE = 1800 (Using the relation CEB = CBE) ...(1)

However,

∠B + ∠CBE = 1800 (Linear pair angles) ...(2)

From equations (1) and (2), we obtain

∠A = ∠B

(ii) AB || CD

∠A + ∠D = 1800 (Angles on the same side of the transversal)

Also,

∠C + ∠B = 1800 (Angles on the same side of the transversal)

∠A + ∠D = ∠C + ∠B

However,

∠A = ∠B [Using the result obtained in (i)

∠C = ∠D

(iii) In ΔABC and ΔBAD,

AB = BA (Common side)

BC = AD (Given)

∠B = ∠A (Proved before)

ΔABC ΔBAD (SAS congruence rule)

(iv) We had observed that,

ΔABC ΔBAD

AC = BD (By CPCT)