Question

in a quadrilateral PQRS,PO and QO are the bisectors of angle P and Q respectively. Prove that angle POQ =½(angle R+S)

Answer 1

Answer:

proved

Step-by-step explanation:

in poq.

poq +1/2 p + 1/2 + q =180

(angle sum property of triangle)

now poq. =180-(1/2p+1/2q)

=180-1/2(p+q)

= 180-1/2{360-(s+r)}

=180. -1/2. *360 +1/2(s+r)

=180-180 +1/2(s+r)

=0+1/2(s+r)

now. poq=1/2(s+r)

hence proved

Answer 2

Given,

PQRS is a quadrilateral.

PO and QO are the bisectors of angles P and W respectively.

To prove,

Angel POQ = ½ (angle R + S)

Solution,

We can easily solve this problem by following the given steps.

Now, we know that the angle bisectors divide the angles into two halves.

According to the question,

PO and QO are the bisectors of angles P and W respectively.

Then, angle P = 2 × angle OPQ

Angle OPQ = angle P/2 ---- equation 1

angle Q = 2× angle OQP

Angle OQP = angle Q/2 ---- equation 2

We know that sum of all the angles in a triangle is 180°.

Now, in ∆ OPQ,

Angle POQ + angle OPQ + angle OQP = 180°

Angle POQ + angle P/2 + angle Q/2 = 180° ( Putting the values from equations 1 and 2)

Taking 1/2 common from the last two angles,

Angle POQ + 1/2 ( angle P + Q) = 180° ---- equation 3

We know that in a quadrilateral the sum of the four angles is 360°.

Angle ( P + Q + R + S) = 360°

Angle (P+Q) = 360- angle (R+S) --- equation 4

Putting the value from equation 4 in equation 3,

Angle POQ + 1/2 [ 360-angle (R+S)] = 180°

Angle POQ + 180° - 1/2 (angle R+S) = 180°

Angle POQ - 1/2 (angle R+S) = 180° - 180° [ Moving 180° from the left-hand side to the right-hand side will result in the change of the sign from plus to minus.]

Angle POQ - 1/2 (angle R+S) = 0

Angle POQ = 1/2 (angle R+S) [ Moving 1/2 (angle R+S) from the left-hand side to the right-hand side will result in the change of sign from minus to plus.]

Hence, Angle POQ = 1/2 (angle R+S) is proved.