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PQRS is a trapezium with PQ∥SR and ∠P=∠Q=50∘ (a) Prove that PS=QR (b) Find ∠S=∠R=?

PQRS is a trapezium with PQ∥SR
and ∠P=∠Q=50∘

(a) Prove that PS=QR

(b) Find ∠S=∠R=?

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 9723 Points
19 days ago
Hint: We solve this problem by constructing the height from vertex S to PQ and R to PQ as follows We use the condition that altitudes between two parallel lines are always equal to prove the two triangles we have as congruent to prove that PS=QR We use the condition that if two triangles are congruent to each other then all corresponding sides and angles are equal to each other. Then we use the condition that the sum of all angles in a quadrilateral is equal to 360∘ so that we can find the required angles. Complete step by step answer: We are given that PQ∥SR and ∠P=∠Q=50∘ Now, let us construct perpendicular from S to PQ and R to PQ as shown in the figure. We know that the condition that altitudes between two parallel lines are always equal By using the above condition we can say that ⇒SM=RN.......equation(i) We know that SM⊥PQ and RN⊥PQ so that the angle at M and N are equal that is ⇒∠SMP=∠RNQ......equation(ii) We are given that the angle at P and Q are equal that is ⇒∠SPM=∠RQN......equation(iii) Now, let us consider the triangles ΔSPM and ΔRQN then we have three condition that from equation (i), equation (ii) and equation (ii) as ⇒SM=RN ⇒∠SMP=∠RNQ ⇒∠SPM=∠RQN Here we can see that two angles and one side are equal So, by using the A.A.S congruence we can say that ΔSPM and ΔRQN are congruent to each other. We know that if two triangles are congruent then all corresponding sides and angles are equal By using the above condition we can conclude that ⇒SP=RQ Hence the required result has been proved. We know that the sum of all angles in a quadrilateral is equal to 360∘ By using the above condition to PQRS we get ⇒∠P+∠Q+∠R+∠S=360∘.....equation(iv) We are given that ∠P=∠Q=50∘ We are asked to find the value ∠S=∠R=? Here we can see that the angles ∠S,∠R are equal to each other. By substituting the required values in equation (iv) we get ⇒50∘+50∘+∠S+∠S=360∘⇒2∠S=360∘−100∘⇒∠S=260∘2=130∘ Therefore we can conclude that ⇒∠S=∠R=130∘ Note: We can find the angles ∠S,∠R in other methods also. We have the standard condition that sum of opposite angles in a quadrilateral is equal to 180∘ By using the above condition to ∠S,∠P then we get ⇒∠S+∠P=180∘ By substituting the required values in above equation we get ⇒∠S+50∘=180∘⇒∠S=130∘ Similarly by applying the above condition to ∠R,∠Q then we get ⇒∠R+∠Q=180∘ By substituting the required values in above equation we get ⇒∠R+50∘=180∘⇒∠R=130∘ Therefore we can conclude that ⇒∠S=∠R=130∘

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