Question

In the adjoining figure, l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that : ΔABC ≅ ΔCDA.

Answer 1

Given: l || m and p || q (Figure).

To Prove: ΔABC ≅ ΔCDA

Proof [Step-wise]:

1. AB is the intercept of line p between l and m and CD is the intercept of line q between l and m.

Since p || q, we have AB || CD.

2. AD lies on line l and BC lies on line m.

Since l || m, we have AD || BC.

3. From (1) and (2) both pairs of opposite sides of quadrilateral ABCD are parallel, so ABCD is a parallelogram.

4. Opposite sides of a parallelogram are equal in length.

Hence AB = CD and AD = BC.

5. AC is common to ΔABC and ΔCDA.

6. Therefore, the three side-pairs are equal:

AB = CD

BC = AD

And AC = CA

By SSS congruence,

ΔABC ≅ ΔCDA.

ΔABC is congruent to ΔCDA.