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.

A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles


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 ≅ ∆RSP

Therefore, ∠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 ≅ ∆ORS

Therefore, 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 = ∠POS

Therefore, ∠POQ = ∠QOR =∠ROS = ∠SOP = 90° (Proved)

17. Opposite angles.






9th Grade Math

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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Successor and Predecessor | Successor of a Whole Number | Predecessor

    May 24, 24 06:42 PM

    Successor and Predecessor of a Whole Number
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Counting Natural Numbers | Definition of Natural Numbers | Counting

    May 24, 24 06:23 PM

    Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

    Read More

  3. Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

    May 24, 24 06:22 PM

    The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

    Read More

  4. Math Questions Answers | Solved Math Questions and Answers | Free Math

    May 24, 24 05:37 PM

    Math Questions Answers
    In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students

    Read More

  5. Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

    May 24, 24 05:09 PM

    Estimating Sum or Difference
    The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

    Read More