Question

Two parallel lines $l$ and $m$ are intersected by another pair of parallel lines $p$ and $q$ as in the figure. Show that triangles $ABC$ and $CDA$ are congruent.

Answer 1

Since $l$ and $m$ are parallel lines and AC is a transversal, $\angle 1=\angle 4$. Similarly transversal AC cuts parallel lines $p$ and $q$, so that $\angle 2=\angle 3$.

In triangles ABC and CDA, we have
$\angle 1=\angle 4$ (proved);
$\angle 2=\angle 3$ (proved);
$AC=AC$ (common side).
By $\text{A.S.A}$ postulate,
$△ABC\cong △CDA$.