# Trapezoid and its Theorems

In this section ,we will discuss some trapezoid and its theorems.

Trapezoid is a quadrilateral with at least one pair of parallel sides. AB || CD. (if there are two pairs of parallel lines then it is a parallelogram)

When non-parallel sides in trapezoid are equal then it is know ans isosceles trapezoid.
Theorem 1: A trapezoid is isosceles if and only if the base angles are congruent. Given : ABCD is an isosceles trapezoid. AD = BC and AB || CD.

Prove that : ∠C = ∠D

 Statements Reasons 1) ABCD is a trapezoid. 1) Given 2) AB || CD 2) Given 3) AD = BC 3) Given 4) DA || CE 4) By construction 5) ADCE is a parallelogram. 5) By Properties of parallelogram. 6) DA = CE and DC = AE 6) By properties of parallelogram. 7) BC = CE 7) BC = AD and AD = CE (Transitive property) 8) ∠CEB ≅ &CBE 8) If BC ≅ CE then angle opposite to them are congruent. 9) ∠DAB ≅ ∠ABC 9) property of parallelogram and linear pair angles 10) ∠A + ∠D = 180 and ∠B + ∠C = 180 10) Interior angles on the same side of the transversal are supplementary. 11) ∠A + ∠D = ∠C + ∠B 11) Transitivity ( Right sides are same so left sides are equal) 12) ∠D = ∠C 12) From above (∠A = ∠B)

Example : In a trapezoid PQRS, PQ||RS and PS = QR. If ∠S = 60 0 then find the remaining angles.
Solution :
PQ||RS and PS = QR, so trapezoid PQRS is an isosceles trapezoid.
In isosceles trapezoid, base angles are equal.(trapezoid and its theorems)
∠S = ∠R and ∠P = ∠Q
But ∠S = 60
0
∴ ∠R = 60
0
Let ∠P = ∠Q = x
Sum of all the angles in a quadrilateral is 360.
∴ ∠P + ∠Q + ∠S + ∠R = 360
x + x + 60 + 60 = 360
2x +120 = 360
2x = 360 -120
2x = 240
∴ x = 240/2
x = 120
∠P = ∠Q = 120
0
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Some important theorems of trapezoids are given below :

 Theorems
 1. A trapezoid is isosceles if and only if the base angles are congruent.
 2. A trapezoid is isosceles if and only if the diagonals are congruent.
 3. If a trapezoid is isosceles, the opposite angles are supplementary.
 The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
 Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

Practice

1) In a trapezoid ABCD,AB|| CD and BC = AD. If m∠C=65
0 then find m∠D.
2) PQRS is a trapezium in which PQ || RS. If ∠P = ∠Q = 40, find the measures of other two angles.
3) In trapezoid ABCD, ∠B= 120
0 Find m∠C.
4) In a quadrilateral HELP, if EP = LH then what type of quadrilateral it is?
5) In a quadrilateral, the angles are in the ratio of 4:5:3:6.Find the measures of each angles.
6) If three angles in the trapezoid are 130
0 ,120 0 ,50 0 and 2x 0 . Find x and the 4th angle.
7) Draw a isosceles trapezoid named PQRS, PS||QR and PQ = SR.