# Bisectors of the Angles of a Parallelogram form a Rectangle

Here we will prove that the bisectors of the angles of a parallelogram form a rectangle.

Given: PQRS is a parallelogram in which PQ ∥ SR and SP ∥ RQ. The bisectors of ∠P, ∠Q, ∠R and ∠S are PJ, QK, RL and SM respectively which enclose the quadrilateral JKLM.

To prove: JKLM is a rectangle.

Proof:

 Statement Reason 1. ∠QPS + ∠PSR = 180°Therefore, $$\frac{1}{2}$$∠QPS + $$\frac{1}{2}$$∠PSR = 90° 1. PQ ∥ SR. 2. ∠SPM + ∠PSM = 90° 2. PJ and SM are bisectors of ∠QPS and ∠PSR respectively. 3. ∠PMS = 90°       ⟹ JM ⊥ ML. 3. Sum of the three angles of ∆PSM is 180°. 4. Taking bisectors of ∠S and ∠R, ML ⊥ LK;Taking bisectors of ∠R and ∠Q, KL ⊥ JK;Taking bisectors of ∠Q and ∠P, JK ⊥ JM. 4. Similarly. 5. JK ∥ ML, JM ∥ KL. 5. Two lines perpendicular to the same line are parallel. 6. JKLM is a parallelogram. (Proved). 6. By statement 5 and one angle, say ∠JML = 90°.

From Bisectors of the Angles of a Parallelogram form a Rectangle 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. ### Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

Feb 22, 24 04:21 PM

How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

2. ### Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

Feb 22, 24 04:15 PM

Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

3. ### Roman Symbols | What are Roman Numbers? | Roman Numeration System

Feb 22, 24 02:30 PM

Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

4. ### Place Value | Place, Place Value and Face Value | Grouping the Digits

Feb 19, 24 11:57 PM

The place value of a digit in a number is the value it holds to be at the place in the number. We know about the place value and face value of a digit and we will learn about it in details. We know th…