CDS 2026 (I) Elementary Mathematics, question 64

Topic: Geometry

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Q64

ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. Which of the following statements is/are correct ?

I. ΔPAD\Delta PAD is similar to ΔPBC\Delta PBC

II. ∠PAD+∠PDA=∠PBC+∠PCB\angle PAD+\angle PDA=\angle PBC+\angle PCB

Select the answer using the code given below :

  1. (a) I only
  2. (b) II onlyCorrect answer
  3. (c) Both I and II
  4. (d) Neither I nor II

Solution.

Answer: (b) II only

  1. In a cyclic quadrilateral an exterior angle equals the interior opposite angle: ∠PCB=∠BAD=∠PAD\angle PCB = \angle BAD = \angle PAD and ∠PBC=∠ADC=∠PDA\angle PBC = \angle ADC = \angle PDA.
  2. With ∠P\angle P common, ΔPAD∼ΔPCB\Delta PAD \sim \Delta PCB in the order P→PP \to P, A→CA \to C, D→BD \to B.
  3. Statement I writes the order PAD→PBCPAD \to PBC, which needs ∠PAD=∠PBC\angle PAD = \angle PBC; that is not true in general, so I is not correct.
  4. Statement II is correct: each sum equals 180∘−∠P180^{\circ} - \angle P.

Trap: The two triangles are similar, so I looks right, but in similarity notation the letter order fixes which angles match, and PBC gives the wrong order.