3 Tutor System
Starting just at 265/hour

# l and m are two parallel lines intersected by another pair of parallel lines p and q (see figure). Show that $$\triangle{ABC}$$ $$\displaystyle \cong$$ $$\triangle{CDA}$$.

From figure, we have,

$$\angle{1}$$ = $$\angle{2}$$ (Vertically opposite angles) ...(i)

$$\angle{1}$$ = $$\angle{6}$$ (Corresponding angles) ...(ii)
$$\angle{6}$$ = $$\angle{4}$$ (Corresponding angles) ...(iii)

From Equations, (i), (ii) and (iii), we have

$$\angle{1}$$ = $$\angle{4}$$ and $$\angle{2}$$ = $$\angle{6}$$ ...(iv)

In $$\triangle{ABC}$$ and $$\triangle{CDA}$$, we have

$$\angle{4}$$ = $$\angle{2}$$ ...(from (iii) and (iv))
$$\angle{5}$$ = $$\angle{3}$$ ...(Alternate angles)

and AC is a common side.

Therefore, we get,
$$\triangle{ABC}$$ $$\displaystyle \cong$$ $$\triangle{CDA}$$ ...(By SAS congruency test)
Hence, proved.