ABCD is a quadrilateral the bisectors of angle A and angle B intersect at P show that angle C+angle D=2 angle APB
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If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, bisectors of ∠B and ∠C at Q, bisectors of ∠C and ∠D at R and bisectors of ∠D and ∠A at S, then PQRS is a
Answer
Correct option is
D
quadrilateral whose opposite angles are supplementary
From question the diagram we get is above.
From figure,
∠QPS=∠APB (Vertically opposite angles) ---- (1)
Consider, △APB
∠APB+
2
1
∠A+
2
1
∠B=180
∘
∠APB=180
∘
−
2
1
(∠A+∠B) ----- (2)
From equation (1) & (2),
∠QPS=180
∘
−
2
1
(∠A+∠B) ----- (3)
Similarly, ∠QRS=180
∘
-
2
1
(∠C+∠D) ---- (4)
From (3) &(4),
∠QPS+∠QRS=360
∘
-
2
1
(∠A+∠B+∠C+∠D)
=360
∘
−
2
1
(360
∘
)
=360
∘
−180
∘
∴ ∠QPS + ∠QRS = 180
∘
Hence, PQRS is a quadrilateral whose opposite angles are supplementary.
Step-by-step explanation: