How to show that a polygon is a trapezoid?

Question

How can I show that $ABCD$ is a trapezoid if all I know is that $AB=8$, $AD=9$, $DC=6$, $BC=16$, and $AC=12$? Can this be solved using similar triangles? I showed that $ABC$ and $DCA$ triangles are similars from (SSS) and $\frac{DC}{AB}=\frac{AD}{AC}=\frac{AC}{BC}=\frac{3}{4}$ how to go from here?

Answer 1

Since $\frac{6}{8}=\frac{CD}{AB}=\frac{9}{12}=\frac{AD}{AC}=\frac{12}{16}=\frac{AC}{BC}$

then $\triangle ACD \sim \triangle CBA$

$\angle CAD$=$\angle ACB$

$\angle ACD$=$\angle B$

$\angle D$=$\angle BAC$

and

$\angle CAD$+$\angle ACB+\angle ACD+\angle B+\angle D+\angle BAC=360^{\circ}$

in the quadrilateral $ABCD$

$\angle A$+$\angle B=180^{\circ}$ (co-interior $\angle$`s are supplementary)

$\angle C$+$\angle D=180^{\circ}$ (co-interior $\angle$`s are supplementary)

Finally, $AD\parallel BC$, $ABCD$ is a trapezium.