$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow ZYX$. If $AB = 3, BC = 5, CA = 6$ and $XY = 6$,find the perimeter of $\Delta XYZ$. (in $.8$)

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow DEF$. If $3 AB = 5 DE$ and $DF = 9$,then $AC = \ldots$

In the figure,the line segment $XY$ is parallel to side $AC$ of $\Delta ABC$ and it divides the triangle into two parts of equal areas. Find the ratio $\frac{AX}{AB}$.

In the figure,if $\Delta ABE \cong \Delta ACD,$ show that $\Delta ADE \sim \Delta ABC.$

In parallelogram $ABCD$,$P$ is a point on side $BC$. The line segment $DP$ intersects the extension of side $AB$ at point $L$. Prove that: $(1) \frac{DP}{PL} = \frac{DC}{BL}$ and $(2) \frac{DL}{DP} = \frac{AL}{DC}$.

Is it true to say that if in two triangles,an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle,then the triangles are similar? Give reasons for your answer.