Question

In the figure, the points M, R, N, S, P and Q are concyclic. Find
Angle PQR + Angle OPR + Angle NMS + Angle OSN, if 'O' is the centre of the circle.​

Answer 1

∠PQR  + ∠OPR + ∠NMS + ∠OSN = 180°.

To find : ∠PQR  + ∠OPR + ∠NMS + ∠OSN = ?

Given :

M, R, N, S, P and Q are concyclic.

"O" is the center of the circle.

From the figure (given),

There are two triangles.

In ΔPQR and ΔMNS,

Radius = OP = OR = OS = OQ = ON  

Hence, ΔPQR and ΔMNS are equilateral triangle.

Angles of equilateral triangle :

∠PQR = 60° and ∠MNS = 60°

In ΔROP,

Angles opposite to sides are equal.

∠OPR = ∠ORP

At center angles subtended twice the angles substended in the circumference.  

(i.e.,) ∠ROP = 2 × ∠RQP

We know the value of ∠PQR = 60° = ∠RQP

So, ∠ROP = 2 × 60° = 120°

      ∠ROP = 120°

In triangle, sum of three angles of triangles is 180°.

In ΔROP,

∠ORP + ∠OPR + ∠ROP = 180°

∠OPR + ∠OPR + 120° = 180°        [ ∵  ∠ORP = ∠OPR ; ∠ROP = 120° ]

2 × ∠OPR = 180° - 120°

2 × ∠OPR = 60°

∠OPR = $\frac{60^{\circ}}{2}$ = 30°

∠OPR = 30°

Similarly in ΔONS,

∠OSN = 30°

Hence,

∠PQR  + ∠OPR + ∠NMS + ∠OSN = 60° + 30° + 60° + 30°

                                                      = 90° + 90°

                                                      = 180°

Therefore, ∠PQR  + ∠OPR + ∠NMS + ∠OSN = 180°.

To learn more...

1. If P, Q and R are mid points of sides BC, CA and AB of a triangle ABC, and Ad is the perpendicular from A on BC, prove that P, Q , R and D are concyclic. plzz solve urgently!!!!

brainly.in/question/1134913

2. ABC is an isosceles triangle in which ab=ac. If angle a=40 then find the value of x and y.​

brainly.in/question/15495133