# 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. ### Comparison of Three-digit Numbers | Arrange 3-digit Numbers |Questions

Sep 13, 24 02:48 AM

What are the rules for the comparison of three-digit numbers? (i) The numbers having less than three digits are always smaller than the numbers having three digits as:

2. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 13, 24 02:23 AM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

3. ### 2nd Grade Place Value | Definition | Explanation | Examples |Worksheet

Sep 13, 24 01:20 AM

The value of a digit in a given number depends on its place or position in the number. This value is called its place value.

4. ### Comparison of Two-digit Numbers | Arrange 2-digit Numbers | Examples

Sep 12, 24 03:07 PM

What are the rules for the comparison of two-digit numbers? We know that a two-digit number is always greater than a single digit number. But, when both the numbers are two-digit numbers