Question

Assertion: If a ray $\overrightarrow{CD}$ stands on a line $\overrightarrow{AB}$, such that ∠ACD = ∠BCD, then ∠ACD = 90°.
Reason: If a ray $\overrightarrow{CD}$ stands on a line $\overrightarrow{AB}$, then ∠ACD + ∠BCD = 180°.
(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.



Assertion:
$$\begin{gathered} \angle ACD+\angle BCD=180°\left[\text{linear pair}\right] \\ \Rightarrow \frac{1}{2}\angle ACD+\frac{1}{2}\angle BCD=90° \\ \Rightarrow \frac{1}{2}\angle ACD+\frac{1}{2}\angle ACD=90°\left[\because \angle BCD=\angle ACD\right] \\ \Rightarrow 2\left(\frac{1}{2}\angle ACD\right)=90° \\ \Rightarrow \angle ACD=\mathbf{90}\mathbf{°} \end{gathered}$$
Hence, the Assertion is true.
Reason:
Clearly, the given statement is true.
Hence, both the Assertion and Reason are true and Reason is a correct explanation of Assertion.