The sum of $3$ distinct angles of a triangle is equal to the sum of $2$ right angles and the difference between two pairs of the angles is $10°$. Find $k$, if the smallest amongst the angles is $k°$.

$P$ is an interior point of a square $ABCD$. Prove that $PA+PB+PC+PD>2AB$.

How can you draw a circle that passes through four vertices of a rectangle? Explain.

In the given figure, $PQRS$ is a parallelogram and $G$ is the centroid of the triangle $PQR$. $A$ is the point of intersection of the diagonals$PR \mathrm{and} SQ$. If $AG=3 \mathrm{cm} \mathrm{and} SQ=k \mathrm{cm}$, find $k$, if the perimeter of the rhombus $ABCD$ is $k \mathrm{cm}$.

In the given figure, $ABCD$ is a square of side $18 \mathrm{cm}$ and $PN\perp BC \mathrm{and} PM\perp AB$.  Find $k$, if the length of $MN$ is $k\sqrt{2} \mathrm{cm}$.

$ABCD$ is a kite in which $AB=AD$ and $CB=CD$. $\angle ABD=30° \mathrm{and} \angle BDC=40°$. Find $k$, if  $\angle A+\angle C=k°$.

Find $k$, if the complement of an angle whose supplement is $100°$ is $k°$.

In the given figure, $AD$ is bisector of $\angle BAC$. Prove that triangles $ABD \mathrm{and} ADC$ are congruent, if $AB=AC$.

$LIVE$ is a rectangle. The diagonal $LV$ is the bisector of the angle $V$. If $IV=2.5 \mathrm{cm}$, then find the length of $LI$. Support your answer with suitable explanation.