IF THE ANGLES P Q R AND S OF A QUADRILATERAL P Q R S ARE IN THE RATIO 11 :19:21:9, then which of the following statements is true? a) pqrs is a parallelogram with PQ||SR. b) pqrs is a trapezium with PQ || SR. c) pqrs is a kite d) pqrs is a trapezium with PS||QR
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IF THE ANGLES P Q R AND S OF A QUADRILATERAL P Q R S ARE IN THE RATIO 11 :19:21:9, then (d) pqrs is a trapezium with PS||QR.
- The angles P,Q,R and S of a quadrilateral PQRS are in the ratio 11:19:21:9.
- Now we obtain the angles of the quadrilateral.
- Sum of all angles of a quadrilateral is equal to 360°.
- Let us consider the common multiple as x.
- Now, the angles P,Q,R and S becomes 11x,19x,21x and 9x.
- Sum of all angles = 360°.
11x + 19x + 21x + 9x = 360
60x = 360
x = 6.
- Angles P,Q,R and S of the quadrilateral becomes 66°,114°,126° and 54°.
- Now, we check from the options given which satisfy the angles obtained.
- First is PQRS is a parallelogram, PQRS cannot be a parallelogram as opposite angles of parallelogram are congruent. But there is no set of congruent angles.
- Now , we construct a rough figure with the angles , we get that there are two set of supplementary angles in the quadrilateral which means the quadrilateral has a set of parallel lines.We have to find out which set of lines are parallel.
- The supplementary angles are ∠P , ∠Q and ∠S , ∠R.
- Therefore, either PS will be parallel to QR or PR will be parallel to QS.
- A quadrilateral which has one set of parallel sides is called a trapezium.
- So the option (4) satisfies the condition as PS║QR.
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