Exercise

One angle of a quadrilateral is of $108°$ and the remaining three angles are equal. If each of the three equal angles are $k^{\circ }$, then find the value of $k$.

$ABCD$ is a trapezium in which $AB \parallel DC$ and $\angle A = \angle B = 45°$. Find angles $C$ and $D$ of the trapezium.

The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $60°$. Find the angles of the
parallelogram.

$\mathrm{ABCD}$ is a rhombus in which altitude from $\mathrm{D}$ to side $\mathrm{AB}$ bisects $\mathrm{AB}$. Find the angles of the rhombus.

Through $\mathrm{A}, \mathrm{B}$ and $\mathrm{C},$ lines $\mathrm{RQ}, \mathrm{PR}$ and $\mathrm{QP}$ have been drawn, respectively parallel to sides $\mathrm{BC}, \mathrm{CA}$ and $\mathrm{AB}$ of a $△\mathrm{ABC}$ as shown in the given figure. Show that $\mathrm{BC}=\frac{1}{2}\mathrm{QR}$.

$\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ are the midpoints of the sides $\mathrm{BC}, \mathrm{CA}$ and $\mathrm{AB}$, respectively of an equilateral triangle $\mathrm{ABC}$. Show that $△\mathrm{DEF}$ is also an equilateral triangle.

Points $\mathrm{P}$ and $\mathrm{Q}$ have been taken on opposite sides $\mathrm{AB}$ and $\mathrm{CD}$, respectively of a parallelogram $\mathrm{ABCD}$ such that $\mathrm{AP}=\mathrm{CQ}$. Show that $\mathrm{AC}$ and $\mathrm{PQ}$ bisect each other.

In the given figure, $\mathrm{P}$ is the midpoint of side $\mathrm{BC}$ of a parallelogram $\mathrm{ABCD}$ such that $\angle \mathrm{BAP}=\angle \mathrm{DAP}$. Prove that $\mathrm{AD}=2\mathrm{CD}$.