There is a slope making an angle of measure $30^{\circ}$ with the ground level. $A$ man walks on the slope for $x \, m$ and reaches a point at a height of $y \, m$ from the ground. Then,which of the following holds true?

$A$ ladder leans against a wall such that its lower end makes an angle of measure $\theta$ with the ground. If its lower end is $a \ m$ away from the base of the wall,then the ladder reaches the wall at a height of $\ldots \ldots \ldots \ldots \ldots \ m$.

An observer $1.5 \ m$ tall is $20.5 \ m$ away from a tower $22 \ m$ high. Determine the angle of elevation of the top of the tower from the eye of the observer. (in $^{\circ}$)

$A$ ladder leans against a wall. It makes an angle of $45^{\circ}$ with the ground and reaches a height of $10 \, m$ on the wall. Find the length of the ladder and the distance between the lower end of the ladder and the base of the wall.

The height of a building is $11 \, m$ and the height of a pole is $8 \, m$. Watching from the midpoint of the line segment joining their bases,the angle of elevation of the top of the pole is $\alpha$ and that of the top of the building is $\beta$. Then,$\ldots \ldots \ldots$ holds true.

In $\Delta ABC$,$\angle B$ is a right angle. If $AC = 20$ and $BC = 10$,then $m\angle ACB = \ldots$ (in $^{\circ}$)