Question 14 Marks

A rectangular MORE is shown below :

Answer the following questions by giving appropriate reason.
(i) Is $R E=O M?$
(ii) Is $\angle MYO =\angle R X E?$
(iii) Is $\angle M O Y=\angle R E X?$
(iv) Is $\triangle M Y O \cong \triangle R X E?$

Answer

(i) Yes, $R E=O M$
Given, the above rectangle, opposite sides are equal.
(ii) Yes, $\angle M Y O=\angle R X E$
Here, $M Y$ and $R X$ are perpendicular to $O E$.
Since, $\angle R X O=90^{\circ} \Rightarrow \angle R X E=90^{\circ}$
and $\angle M Y E=90^{\circ} \Rightarrow \angle M Y O=90^{\circ}$
(iii) Yes, $\angle M O Y=\angle R E X\quad$ [Since, these are alternate interior angles, as $R E \| O M$ and $E O$ is a transversal]
(iv) Yes, $\triangle M Y O \cong \triangle R X E$
In $\triangle M Y O$ and $\triangle R X E$, we see that
$M O=R E \qquad\qquad\qquad\text{ [proved]}$
$ \angle M O Y=\angle R E X \qquad\quad\text{ [proved]}$
$ \angle M Y O=\angle R X E \qquad\quad\text{ [proved]}$
$ \therefore \triangle M Y O \cong \triangle R X E\quad\quad\text{[by AAS]}$

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