The angle between lines $L$ and $M$ measures ${35}^o$ degrees. If line $M$ is rotated ${45}^o$ degrees counter clockwise about point $P$ to line $M^1$ what is the angle in degrees between lines $L$ and $M^1$

In the figure given below, $XYZ$ is a right angled triangle in which $\angle Y={45}^o$ and $\angle X={90}^o$. $ABCD$ is a square inscribed in it whose area is $64 c{m}^2$. What is the area of triangle $XYZ$?

Find the area of the shaded portion

In the figure $ABDC$ is a cyclic quadrilateral with $O$ as center of the circle. Find $\angle BDC$.

$B,O,P$ are centers of semicircles $AXC,AYB,$ and  $BZC$ respectively. $AC=12 cm$. Find the area of the shaded region.

$O$ is the centre of the circle. $OP=5$ and $OT=4$, and $AB=8$. The line $PT$ is a tangent to the circle. Find $PB$

In the figure given below, $AB=16, CD=12$ and $OM=6$. Calculate $ON$.