Exam Practice

In rectangle $DATE$, diagonals $DT$ ard $AE$ intersect at $S$. If $AE=40$ and $ST=x+5$, then find the value of $x$.

In a parallelogram $ABCD$, diagonals $AC$ and $BD$ intersect at $O$ and $AC=6.8 \mathrm{cm}$ and $BD=5.6 \mathrm{cm}$. Find the measures of $OC$ and $OD$.

In $∆ABC, D, E$ and $F$ are the mid-points of the sides $AB, BC$ and $AC$, respectively. Then, prove that quadrilateral $DECF$ is a parallelogram.

The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ intersect each other at the point $O$. If $\angle DAC=42°$ and $\angle AOB=90°$, then find the value of $\angle DBC$.

$ABCD$ is a parallelogram in which $\angle ADC=75°$ and side $AB$ is produced to point $E$ as shown in the figure. Find $(x+y)$.

If the bisectors of two adjacent angles $A$ and $B$ of a quadrilateral $ABCD$ intersect at a point $O$, such that $\angle C+\angle D=k\angle AOB$, then find the value of $k$.

The sides $BA$ and $DC$ of a quadrilateral $ABCD$ are produced as shown in figure. Prove that $a+b=x+y$.

Points $P$ and $Q$ have been taken on opposite sides $AB$ and $CD$ respectively of a parallelogram $ABCD$ such that $AP=CQ$ (see figure). Show that $AC$ and $PQ$ bisect each other.