In given fig.

Rays OA, OB, OC, OD and OE have common initial point O. Show that `angleAOB + angleBOC + angleCOD + angleDOE + angleEOA = 360^@`.

In Fig, OA . OB = OC . OD. Show that angle A=angle C and anlge B=angleD .

In fig. OD is the bisector of angleAOC , OE is the bisector of angleBOC and ODbotOE . Show that the points A,O and B are collinear

In Fig. 7.8, OA = OB and OD = OC. Show that (i) Delta AOD cong Delta BOC and (ii) AD || BC.

O is any point inside a rectangle ABCD (see Fig. 6.52). Prove that OB^2 + OD^2 = OA^2 + OC^2 .

ABCD is a parallelogram and P the intersection of the diagonals, O is any point . Show that vec(OA)+vec(OB)+vec(OC)+vec(OD)=4vec(OP) .

In fig., O is a point in the interior of a triangle ABC, OD bot BC, OE bot AC and OF bot AB. Show that:- OA^2+OB^2+OC^2-OD^2-OE^2-OF^2=AF^2+BD^2+CE^2 .

ABCD is a quadrilateral and O is point in its plane. Show that if vec OA+ vec OB+ vec OC+ vec OD= vec 0 , then O is the point of the interection of the lines joining the mid-points of the opposite sides of ABCD.

Here is a ray vec(OA) . It starts at O and passes through the point A. It also passes through the point B. Can you also name it as vec(OB) ? Why? vec(OA) and vec(OB) are same here. Can we write vec(OA) as vec(AO) ? Why or why not? Draw five rays and write appropriate names for them. , What do the arrows on each of these rays show?

In fig., O is a point in the interior of a triangle ABC, OD bot BC, OE bot AC and OF bot AB. Show that:- AF^2+BD^2+CE^2=AE^2+CD^2+BF^2 . .

Given three non-coplanar vectors OA=a, OB=b, OC=c. Let S be the centre of the sphere passing through the points O, A, B, C if OS=x, then