# ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that(i) ∠ A = ∠ B(ii) ∠ C = ∠ D(iii) Δ ABC ≅Δ BAD(iv) Diagonal AC = diagonal BD[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

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

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,

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

AB = BA (Common side)

B = A (Proved before)

AC = BD (By CPCT)

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Critical Thinking Problems on Quadrilaterals44 mins
Quiz | Properties of Parallelogram31 mins
Quiz | Basics of Quadrilaterals36 mins
RD Sharma | Extra Qs. of Cyclic Quadrilaterals31 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses