Question

Assertion: ∆ABC and ∆DBC are two isosceles triangles on the same base BC. Then, ∠ABD = ∠ACD.

Reason: The angles opposite to equal sides of a triangle are equal.
(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

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

We know that the angles opposite to the equal sides of a triangle are equal.
So, the reason is true.

$$\begin{gathered} \mathrm{AB}=\mathrm{AC}\Rightarrow \angle \mathrm{ABC}=\angle \mathrm{ACB} \\ \mathrm{DB}=\mathrm{DC}\Rightarrow \angle \mathrm{DBC}=\angle \mathrm{DCB} \\ \mathrm{Subtracting}\mathrm{them},\mathrm{we}\mathrm{get}: \\ \Rightarrow \angle \mathrm{ABC}-\angle \mathrm{DBC}=\angle \mathrm{ACB}-\angle \mathrm{DCB} \\ \therefore \angle \mathrm{ABD}=\angle \mathrm{ACD} \end{gathered}$$

So, the assertion (A) is true and, clearly, the reason (R) gives the assertion.