In the given figure, the isosceles triangle $PQR\cong TPS$. The point $T$ is equidistant from both $P$ and $Q$.

 

What is the value of $x$?

In the figure below, $O$ is the centre of the circle. $A$ and $B$ are points on the circle. $\angle OAP$ and $\angle OBP$ are right angles.

Which of the following congruence rules can we use to prove $PA=PB$?

Shown below are some triangles which Salim constructed: $△OAB, △OAC, \mathrm{and} △OAD$

Now he wants to construct one more triangle $△OAE$, such that $OA+OE=AE$.

He marks point $E$ on the circle and joined $E$ with points $A$ and $O$.

Which of the following constructions could he have made?

In the figure below, $QM$is the median of $∆PQR$. $QM$is extended to $S$ to get $QS$. $PT\perp QM$, where $T$ is a point on $QM$ and $RS\perp QS$.

Which of these is congruence rule(s) can we apply to prove $PT=RS$?

In a triangle with integer side lengths, one side is three times as long as the second side and the length of the third side is $17$ units. What is the greatest possible perimeter of the triangle.

The three side lengths of a non-degenerate triangle are $x+3, x+7$ and $x+16$. Which of the following is NOT a possible value for $x?$

In $△ABC$ above, $\angle A=66°, M$ is the midpoint of side $\overline{BC}$, and $D$ and $E$ are the feet of perpendicular drawn from $M$ to $\overline{AB}$ and  $\overline{AC}$, respectively. If $\left|\overline{MD}\right|=\left|\overline{ME}\right|$, what is the measure of $\angle CME$?