# A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles

Here we will prove that a rhombus is a parallelogram whose diagonals meet at right angles.

Given: PQRS is a rhombus. So, by definition,

PQ = QR = RD = SP. Its diagonals PR and QS intersect at O.

To prove: (i) PQRS is a parallelogram.

(ii) ∠POQ = ∠QOR = ∠ROS = ∠SOP = 90°.

Proof:

 Statement Reason (i) In ∆PQR and ∆RSP,1. PQ = RS and QR = PS 1. Given. 2. PR = RP 2. Common side 3. ∆PQR ≅ ∆RSPTherefore, ∠QPR = ∠SRP, ∠QRP = ∠SPR. 3. By SSS criterion of congruency. CPCTC 4. SR ∥ PQ, PS ∥QR. 4. Alternate angles are equal. 5. PQRS is a parallelogram. (Proved)(ii) In ∆OPQ and ∆ORS, 5. By definition. 6. ∠OPQ = ∠ORS 6. By statement 4, PQ ∥ SR and PR is a transversal. 7. ∠OQP = ∠OSR 7. P PQ ∥ SR and QS is a transversal 8. PQ = SR 8. Given. 9. ∆OPQ ≅ ∆ORSTherefore, OP = OR, OQ= OS.In ∆POS ≅ ∆ROS, 9. By AAS criterion of congruency. CPCTC 10. PS = RS 10. Given. 11. OP = OR 11. From statement 10. 12. OS = SO 12. Common side. 13. Therefore, ∆POS ≅ ∆ROS 13. By SSS criterion of congruency. 14. ∠POS = ∠ROS 14. CPCTC 15. ∠POS + ∠ROS = 180° 15. Linear pair. 16. ∠POS = ∠ROS = 90° 16. From statements 14 and 15. 17. ∠POQ = ∠ROS, ∠QOR = ∠POSTherefore, ∠POQ = ∠QOR =∠ROS = ∠SOP = 90° (Proved) 17. Opposite angles.

From A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles 1. ### 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

Dec 06, 23 01:23 AM

2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

2. ### Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

Dec 04, 23 02:14 PM

Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes: