How many triangles can be drawn having its angles as $53^{\circ}, 64^{\circ}$ and $63^{\circ}$? Give reason for your answer.

The image of an object placed at a point $A$ before a plane mirror $LM$ is seen at the point $B$ by an observer at $D$ as shown in the figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.

$AB$ and $CD$ are the smallest and largest sides of a quadrilateral $ABCD$. Determine which is greater,$\angle B$ or $\angle D$.

State whether each of the following statements is true or false:
$(1)$ If a transversal intersects two parallel lines,then the interior angles on the same side of the transversal are equal.
$(2)$ The maximum number of acute angles in any triangle is two.

$ABC$ is a right triangle such that $AB = AC$ and the bisector of angle $C$ intersects the side $AB$ at $D$. Prove that $AC + AD = BC$.

In triangles $ABC$ and $PQR$,$AB = AC$,$\angle C = \angle P$ and $\angle B = \angle Q$. The two triangles are