# ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.

**Solution:**

Given, ABCD is a __quadrilateral__

AB || DC

AD = BC

We have to prove that ∠A = ∠B and ∠C = ∠D.

Extend AB to E and join CE such that AD || CE

So, AECD is a __parallelogram__.

We know that the opposite sides of a parallelogram are parallel and __congruent__.

AD = EC

Given, AD = BC

So, BC = EC

We know that the angles opposite to equal sides are equal.

∠CBE = ∠CEB ----------- (1)

We know that the __linear pair of angles__ is always supplementary.

∠B + ∠CBE = 180° -------- (2)

Since AD || EC and cut by transversal AE,

We know that two parallel lines are cut by a __transversal__, the sum of interior angles lying on the same side of the transversal is always supplementary.

∠A + ∠CEB = 180° --------- (3)

From (1),

∠A + ∠CEB = 180° ------------- (4)

Comparing (2) and (4),

∠A = ∠B

We know that the sum of supplementary angles of a parallelogram is always equal to 180 degrees.

∠A + ∠D = 180°

∠B + ∠C = 180°

On comparing,

∠A + ∠D = ∠B + ∠C

since , ∠A = ∠B

∠A + ∠D = ∠A + ∠C

∠C = ∠D

Therefore, it is proven that ∠A = ∠B and ∠C and ∠D

**✦ Try This: **In a trapezium ABCD with AB parallel to CD, the diagonals intersect at P. The area of △ABP is 72 square cm and of ΔCDP is 50 square cm. Find the area of the trapezium.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 8**

## ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.

**Summary:**

A parallelogram is a two-dimensional geometrical shape, whose sides are parallel to each other and equal in length. ABCD is a quadrilateral in which AB || DC and AD = BC. It is proven that ∠A = ∠B and ∠C = ∠D

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