State the theorem which justifies the conclusion

If  $\overline{OM}\equiv \overline{OT} ; \overline{OM} \perp \overline{RS}; \overline{OT} \perp \overline{PQ}$ and $\overline{OM}\equiv \overline{OT}$ then $\overline{PQ}\cong \overline{RS}$

 

State the theorem which justifies the conclusion

If  $\overline{AB}\equiv \overline{CD} ; \overline{PM} \perp \overline{AB}$ and $\overline{PN} \perp \overline{CD}$, prove  $\overline{PM}\cong \overline{PN}$

The diameter and a chord of a circle have a common endpoint as shown in the figure. If the length of the diameter is $40 \mathrm{cm}$, the length of the chord is $24 \mathrm{cm}$ and the distance of midpoint of the chord from the centre is $k \mathrm{cm}$ then find $k$.

A chord whose midpoint is $15 \mathrm{cm}$away from the centre of the circle is $16 \mathrm{cm}$ long. If the radius of the circle is $r \mathrm{cm}$, then find the value of $r$.

Given a circle with centre $O$ and diameter $PQ. AQ$ and $BQ$ are chords such that $\angle AQP$ and $\angle BQP$ are congruent. Prove that chord $AQ$is congruent to chord $BQ.$

In a circle with centre $O$ chords $AB, BC$ and $AC$ are drawn. Perpendicular bisectors $\overline{PO}, \overline{OQ}$ and $\overline{OR}$ are concurrent and are equal. Prove $△ABC$  is equilateral. 

Given a circle with centre $P. P$ is the midpoint of $MN$. $MN$ is perpendicular to $\overline{BC}$  as well as $\overline{AD}$. Prove that  $\overline{ABCD}$ is a rectangle.

Find the angle marked $x$ in the following figure

$\angle x =$ _____

In the above figure, $\angle x =$_____$°$.