Assertion (A): Using the information in the given figure, we get AM = CN. Reason (R): ∠NDC = 1/2 ∠ADC and, ∠MBC = 1/2 ∠ABC Since. ∠ADC = ∠ABC ⇒ ∠NDC = ∠MBC ⇒ ∠AM = CN 1. A is true, R is false. 2. A is false, R is true. 3. Both A and R are true. 4. Both A and R are false.

Mathematics

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A is true, R is false.

Explanation

In Δ AMB and Δ CDN,

∠ MAB = ∠ NCD (alternate interior angles as AC is a transversal)

∠NDC = ∠MBC (∠NDC = $\dfrac{1}{2}$ ∠ADC and, ∠MBC = $\dfrac{1}{2}$ ∠ABC Since. ∠ADC = ∠ABC ⇒ ∠NDC = ∠MBC)

AB = CD (Opposite sides of parallelogram are always equal)

So, using the ASA Congruency criterion,

Δ AMB ≅ Δ CDN

Hence, their corresponding sides are equal i.e, AM = CN.

∴ Assertion (A) is true.

∠NDC = $\dfrac{1}{2}$ ∠ADC
and, ∠MBC = $\dfrac{1}{2}$ ∠ABC
Since. ∠ADC = ∠ABC

⇒ ∠NDC = ∠MBC

But, ∠NDC = ∠MBC ≠ ∠AM = CN

∴ Reason (R) is false.

Hence, Assertion (A) is true, Reason (R) is false.

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