Using ruler and compasses only, construct a parallelogram ABCD with AB = 5 cm, AD = 2.5 cm and ∠BAD = 45°. If the bisector of ∠BAD meets DC at E, prove that ∠AEB is a right angle.

Steps of construction:

1. Draw AB = 5.0 cm.
2. At A, construct ∠BAP = 45°.
3. With A as centre and radius 2.5 cm cut the line AP at D.
4. With D as centre and radius 5.0 cm, draw an arc.
5. With B as center and radius 2.5 cm, draw an arc to meet the previous arc at C.
6. Join BC and CD. Then, ABCD is the required parallelogram

Draw the bisector of ∠BAD, which cuts DC at E and join EB.

On measuring ∠AEB, it is equal to 90°.

Hence, proved that ∠AEB is a right angle.