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If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠C and ∠D at R and of ∠D and ∠A at S, then PQRS is a
- a)Rectangle
- b)Quadrilateral whose opposite angles are supplementary
- c)Parallelogram
- d)Rhombus
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If bisectors of∠Aand∠Bof a quadrilateral ABCD intersect each o...
To show: ∠PSR + ∠PQR = 180°
∠SPQ + ∠SRQ = 180°
∠SPQ + ∠SRQ = 180°
In △DSA,
∠DAS + ∠ADS + ∠DSA = 180° (angle sum property)
+ ∠ SA = 180° (since RD and AP are bisectors of ∠D and ∠A)
∠DSA = 180°
∠PSR = 180°−
∠PSR = 180°−
(∵ ∠DSA = ∠PSR are vertically opposite angles)
Similarly,
∠PQR = 180°− 
Adding (i) and (ii), we get, ∠PSR + ∠PQR = 180°
=360° − 1/2 × (∠A + ∠B + ∠C + ∠D)
=360°− 1/2 × 360° = 180° ∴ ∠PSR + ∠PQR = 180°
In quadrilateral PQRS,
∠SPQ + ∠SRQ + ∠PSR + ∠PQR = 360°
=> ∠SPQ + ∠SRQ + 180° = 360°
=> ∠SPQ + ∠SRQ = 180°
=> ∠SPQ + ∠SRQ + 180° = 360°
=> ∠SPQ + ∠SRQ = 180°
Hence, showed that opposite angles of PQRS are supplementary.
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Understanding the Problem
In the given quadrilateral ABCD, the angle bisectors of angles A, B, C, and D intersect at points P, Q, R, and S respectively. We need to analyze the shape formed by these intersection points.
Angle Bisectors and Their Properties
- Angle Bisector Definition: An angle bisector divides an angle into two equal halves.
- Intersecting Angle Bisectors: The points where these bisectors intersect create a new quadrilateral (PQRS).
Supplementary Angles in Quadrilaterals
- Sum of Angles: In any quadrilateral, the sum of the interior angles is 360 degrees.
- Angle Relationships: When considering angle bisectors, the angles formed at points P, Q, R, and S relate to the original angles of quadrilateral ABCD.
Opposite Angles of PQRS
- Property of Opposite Angles: By the nature of angle bisectors, the angles at points P and R are formed by the halves of the angles at A and C, while Q and S are formed by the halves of angles at B and D.
- Supplementary Angles: This arrangement leads to the conclusion that opposite angles in quadrilateral PQRS are supplementary, meaning they add up to 180 degrees.
Conclusion
Given that the opposite angles of quadrilateral PQRS are supplementary, we conclude that:
- PQRS is a special type of quadrilateral where opposite angles are supplementary.
Therefore, the correct answer is option B: Quadrilateral whose opposite angles are supplementary.
In the given quadrilateral ABCD, the angle bisectors of angles A, B, C, and D intersect at points P, Q, R, and S respectively. We need to analyze the shape formed by these intersection points.
Angle Bisectors and Their Properties
- Angle Bisector Definition: An angle bisector divides an angle into two equal halves.
- Intersecting Angle Bisectors: The points where these bisectors intersect create a new quadrilateral (PQRS).
Supplementary Angles in Quadrilaterals
- Sum of Angles: In any quadrilateral, the sum of the interior angles is 360 degrees.
- Angle Relationships: When considering angle bisectors, the angles formed at points P, Q, R, and S relate to the original angles of quadrilateral ABCD.
Opposite Angles of PQRS
- Property of Opposite Angles: By the nature of angle bisectors, the angles at points P and R are formed by the halves of the angles at A and C, while Q and S are formed by the halves of angles at B and D.
- Supplementary Angles: This arrangement leads to the conclusion that opposite angles in quadrilateral PQRS are supplementary, meaning they add up to 180 degrees.
Conclusion
Given that the opposite angles of quadrilateral PQRS are supplementary, we conclude that:
- PQRS is a special type of quadrilateral where opposite angles are supplementary.
Therefore, the correct answer is option B: Quadrilateral whose opposite angles are supplementary.
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