CDS 2026 (I) Elementary Mathematics, question 72

Topic: Geometry

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Q72

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question : ABCD is a quadrilateral in which AD=BCAD=BC. What is ∠ABC+∠ADC\angle ABC+\angle ADC equal to ?

Statement-I : AB is parallel to DC

Statement-II : AD is not parallel to BC

Which one of the following is correct in respect of the above Question and the Statements ?

  1. (a) The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
  2. (b) The Question can be answered by using either Statement alone
  3. (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement aloneCorrect answer
  4. (d) The Question cannot be answered even by using both the Statements together

Solution.

Answer: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

  1. I alone allows a parallelogram. A non-rectangular parallelogram can have ∠B=∠D=60∘\angle B = \angle D = 60^{\circ}, giving a sum of 120°, whereas a rectangle gives 180°.
  2. Both have AD=BCAD = BC, so I alone is insufficient. II alone permits both a proper isosceles trapezium, with opposite-angle sum 180°, and a noncyclic quadrilateral.
  3. For a concrete noncyclic example, take A=(0,0),B=(4,0),C=(1,4),D=(0,5)A = (0,0),B = (4,0),C = (1,4),D = (0,5). Then AD=BC=5AD = BC = 5 and AD is not parallel to BC, while ∠B\angle B ≈ 53.13° and ∠D=45∘\angle D = 45^{\circ}.
  4. Insufficient. Together, AB∥DCAB \parallel DC and AD not parallel to BC exclude the parallelogram case. Equal legs AD=BCAD = BC then give a proper isosceles trapezium, which is cyclic.
  5. Thus ∠ABC+∠ADC=180∘\angle ABC + \angle ADC = 180^{\circ} uniquely.