Question

In the given figure, PQRS is a rhombus in which the diagonal PR is produced to T. If ∠SRT = 152°, find x, y and z.

Answer 1

Rhombus PQRS is given.

Diagonal PR is produced to T.

Also, ∠SRT = 152°.

We know that in a rhombus, the diagonals bisect each other at right angle.

Therefore,

 y = 90°

Now,

∠1 + ∠SRT = 180°

 ∠1 +152° = 180°

           ∠1 = 28°

In ΔSOR, by angle sum property of a triangle, we get:

    ∠1 +y +∠OSR = 180°

28° +90° +∠OSR = 180°

       118° +∠OSR = 180°

                  ∠OSR = 62°

Or,  ∠QSR = 62° (Because O lies on SQ)

We have, SP || PQ .Thus the alternate interior opposite angles must be equal.

Therefore,

x = ∠QSR

x = 62°

In ΔSPR,we have

Since opposite sides of a rhombus are equal.

Therefore,

PS = SR

Also,

Angles opposite to equal sides are equal.

Thus,

 z = ∠1

But ∠1 = 28°

Thus,  z = 20°

Hence the required values for x,y and z are 62°,90°  and 28° respectively.