# 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.

Quadrilateral

Introduction to Quadrilateral
Types of Quadrilateral
Properties of Quadrilateral
Parallelogram and its Theorems
Rectangle and its Theorems
Square and its Theorems
Rhombus and its Theorems
Trapezoid and its Theorems
Kite and its Theorems
Mid Point Theorem

Geometry

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