Question

Assertion: In the given figure, ∆ABC is given with AB = AC and BA is produced to D such that AB = AD. Then, ∠BCD = 90°.

Reason: In the given figure AB = AC and D is a point on BC produced. Then, AB > AD.

(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

(c) Assertion is true and Reason is false.
Reason (R):
$$\begin{gathered} AB=AC \\ \Rightarrow \angle ACB=\angle ABC...\left(i\right) \end{gathered}$$
The side BC of triangle ABC is produced to D.
$$\begin{gathered} \angle ACD>\angle ACB=\angle ABC \\ \mathrm{But}\mathrm{in}∆\mathrm{ADC},\mathrm{side}\mathrm{DC}\mathrm{is}\mathrm{produced}\mathrm{to}\mathrm{B}. \\ \angle ACB>\angle ADC...\left(ii\right) \\ \Rightarrow \angle ABC>\angle ADC[\mathrm{From}(\mathrm{i})\mathrm{and}(\mathrm{ii}\left)\right] \\ \Rightarrow AD>AB \end{gathered}$$
Hence, the reason (R) is false.
Clearly, the assertion is true, but the reason (R) is false.