In the adjoining figure, QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR. If XS ⊥ QR and XT ⊥ PQ; prove that :

(i) △ XTQ ≅ △ XSQ

(ii) PX bisects angle P.

In the following figures, the sides AB and BC and the median AD of the triangle ABC are respectively equal to the sides PQ and QR and median PS of the triangle PQR. Prove that △ ABC and △ PQR are congruent.

The following figure shows a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN. Prove that :

(i) AM = AN

(ii) △ AMC ≅ △ ANB

(iii) BN = CM

(iv) △ BMC ≅ △ CNB

In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE.