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Is The Base Angles Of A Trapezoid Are _______ Congruent, Is The Base Angles Of A Trapezoid Congruent

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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

StatementsReasons1) ABCD is a trapezoid.1) Given2) AB || CD2) Given3) AD = BC3) Given4) DA || CE4) By construction5) ADCE is a parallelogram.5) By Properties of parallelogram.6) DA = CE and DC = AE6) By properties of parallelogram.7) BC = CE7) BC = AD and AD = CE (Transitive property)8) ∠CEB ≅ &CBE8) If BC ≅ CE then angle opposite to them are congruent.9) ∠DAB ≅ ∠ABC9) property of parallelogram and linear pair angles10) ∠A + ∠D = 180 and ∠B + ∠C = 18010) Interior angles on the same side of the transversal are supplementary.11) ∠A + ∠D = ∠C + ∠B11) Transitivity ( Right sides are same so left sides are equal)12) ∠D = ∠C12) From above (∠A = ∠B)

Example : In a trapezoid PQRS, PQ||RS and PS = QR. If ∠S = 600 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 = ∠QBut ∠S = 600∴ ∠R = 600Let ∠P = ∠Q = xSum of all the angles in a quadrilateral is 360.∴ ∠P + ∠Q + ∠S + ∠R = 360x + x + 60 + 60 = 3602x +120 = 3602x = 360 -1202x = 240∴ x = 240/2x = 120∠P = ∠Q = 1200__________________________________________________________________Some important theorems of trapezoids are given below :

Theorems
1. A trapezoid is isosceles if and only if the base angles are congruent.

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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.

Practice1) In a trapezoid ABCD,AB|| CD and BC = AD. If m∠C=650 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= 1200 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 1300,1200,500 and 2×0. 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 TheoremGeometryHome Page

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