Question

Assertion (A): In parallelogram ABCD, PD bisects ∠ADC and PC bisects angle ∠DCB; then ∠DPC = 90°.

Reason (R): `∠PDC = 1/2 xx ∠ADC`,

`∠PDC = 1/2 xx ∠BCD`,

`∠PDC + ∠PDC = 1/2 xx (∠ADC + ∠BCD)`

पर्याय

• A is true, R is false.

• A is false, R is true.

• Both A and R are true, and R is the correct reason for A.

• Both A and R are true, and R is the incorrect reason for A.

Answer 1

Both A and R are true, and R is the correct reason for A.

Explanation:

• Assertion (A): In parallelogram ABCD, consecutive angles ∠ADC and ∠BCD are supplementary, meaning their sum is 180°.
  Since PD and PC bisect these angles respectively, we have ∠PDC = `1/2` ∠ADC and ∠PCD = `1/2` ∠BCD.
  Adding them gives ∠PDC + ∠PCD = `1/2`(∠ADC + ∠BCD) = `1/2` (180°) = 90°.
  In triangle PDC, the sum of angles is 180°, so ∠DPC = 180° − (∠PDC + ∠PCD) = 180° − 90° = 90°.
  Thus, Assertion (A) is true.
• Reason (R): The statements provided under Reason (R) correctly show that:
  ∠PDC = `1/2` × ∠ADC, ∠PCD = `1/2` × ∠BCD, and ∠PDC + ∠PCD = `1/2` × (∠ADC + ∠BCD), which mathematically justifies the derivation leading to the measure of ∠DPC.
  Thus, Reason (R) is true and correctly explains Assertion (A).