Question

Assertion: In the given figure, ∠ABC = 70° and ∠ACB = 30°. Then, ∠BDC = 70°.

Reason: In the given figure, ∠AOC = 130°, then ∠ABC = 115°.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(d) The assertion is false and the reason is true.

Reason (R):
We know that $\angle \mathrm{ADC}=\frac{1}{2}\angle \mathrm{AOC}=\left(\frac{1}{2}\times 130°\right)=65°$
Also, ∠ABC + ∠ADC = 180° (Opposite angles of a cyclic quadrilateral)
⇒ ∠ABC = (180° - 65°) = 115°, which is true.
∴ Reason (R) is true.

Assertion(A):
∠ABC + ∠BAC + ∠BCA = 180°
⇒ 70° + ∠BAC + 30° = 180°
⇒ ∠BAC = (180° - 100°) = 80°
∴ ∠BDC = ∠BAC = 80° (Angles in the same segment)
This is false.
Thus, assertion (A) is false and reason (R) is true.