Complete the  star shaped rangolis by filling them with as many equilateral triangles of side $1$ cm as you can. Count the number of triangles in each case. Which has more triangles?
 

In quadrilateral $ACBD, AC=AD$ and $AB$ bisects $\angle A$ (see given figure). Show that $△ABC\cong △ABD.$ What can you say about $BC$ and $BD?$

$ABCD$ is a quadrilateral in which $AD=BC$ and $\mathrm{\angle }DAB=\mathrm{\angle }CBA$ (see given figure). Prove that $△ABD\cong △BAC$.

$ABCD$ is a quadrilateral in which $AD=BC$ and $\mathrm{\angle }DAB=\mathrm{\angle }CBA$ (see given figure). Prove that $BD=AC$.

$ABCD$ is a quadrilateral in which $AD=BC$ and $\mathrm{\angle }DAB=\mathrm{\angle }CBA$ (see given figure). Prove that $\mathrm{\angle }ABD=\mathrm{\angle }BAC$.

$AD$ and $BC$ are equal perpendiculars to a line segment $AB$ (see given figure). Show that $CD$ bisects $AB\mathit{.}$

$l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$ (see given figure). Show that $△ABC\cong △CDA.$

Line $l$ is the bisector of an angle $\mathrm{\angle }A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\mathrm{\angle }A$. Show that$△APB\cong △AQB$.