In □ABCD, seg AD || seg BC. Diagonal AC and diagonal BD intersect each other in point P. Then show that

In ΔPQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR.

Complete the proof by filling in the boxes. In ΔPMQ, ray MX is bisector of ∠PMQ.

.......... (I) theorem of angle bisector.

In ΔPMR, ray MY is bisector of ∠PMR.

.......... (II) theorem of angle bisector.

But .......... M is the midpoint QR, hence MQ = MR.

∴ XY || QR .......... converse of basic proportionality theorem.

In fig 1.78, bisectors of ∠B and ∠C of ΔABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find

In fig 1.80, XY || seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC.

Activity :

.......... by componendo.

.......... (I)

.......... test of similarity.

.......... corresponding sides of similar triangles.

...from (I)

In figure 1.81, the vertices of square DEFG are on the sides of ΔABC, ∠A = 90°. Then prove that DE² = BD × EC

(Hint : Show that ΔGBD is similar to ΔDFE. Use GD = FE = DE.)