Watching from the top of a $150\, m$ high tower,the angles of depression of two houses situated in the east and west of the tower are found to be $30^{\circ}$ and $45^{\circ}$ respectively. Find the distance between the houses in meters.

Observing from the top of a building,the angle of elevation of the top of a tower is $30^{\circ}$ and the angle of depression of the base of the tower is $60^{\circ}$. If the height of the building is $36 \, m$,find the height of the tower (in $m$).

$A$ flag-staff stands on the top of a building. The angle of elevation of the top of the building is $45^{\circ}$ and that of the top of the flag-staff is $60^{\circ}$,as observed by a man standing $15\, m$ away from the building. Find the height of the flag-staff from the top of the building (in $m$).

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.

The angle of elevation of the top of a tower from a point on the ground,which is $30\, m$ away from the foot of the tower,is $30^{\circ}$. Find the height of the tower.

Two poles are $x$ metres apart and the height of one is double than that of the other. If from the midpoint of the line joining their feet,an observer finds the angle of elevation of their tops to be complementary,then the height of the shorter pole is $\ldots \ldots \ldots \ldots$