Properties of Rectangles

Key Concepts

• Identify a rectangle.
• Explain the conditions required for a parallelogram to be a rectangle.

Rectangle 

A parallelogram in which each pair of adjacent sides is perpendicular is called a rectangle. 

Theorem 

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 

Given: AC=BD

To prove: ABCD is a rectangle. 

Proof:  

Let the sides of the radio be AB, BC, CD and AD

Here,

AB ∥ CD and AD∥BC

Since the opposite sides of the radio are parallel, so, it is in the shape of a parallelogram. 

Now, in △ABC and △DCB,

AB=CD [Opposite sides of a parallelogram] 

BC=BC [Reflexive property] 

AC=BD [Given] 

So, △ABC≅ △DCB by Side-Side-Side congruence criterion. 

Then, ∠ABC=∠DCB∠ABC=∠DCB [Congruent parts of congruent triangles] 

We know that consecutive angles of a parallelogram are supplementary. 

So,

∠ABC+∠DCB=180°

∠ABC+∠ABC=180°

2 ∠ABC=180°

∠ABC=90°

Therefore,

∠DCB=90°

We know that the opposite angles of a parallelogram are equal. 

So, in parallelogram

ABCD, ∠A=∠C=90° and ∠B=∠D=90°∠B=∠D=90°

A parallelogram who’s all the angles measure 90° is a rectangle. 

Theorem 

If a parallelogram is a rectangle, then its diagonals are congruent. 

Given:

∠PQR=∠QRS=∠RSP=∠SPQ=90°

To prove: PR=QS

Proof: Let PQRS be a rectangle. 

In △QPS and △RSP

QP=RS [Opposite sides of a rectangle are equal] 

PS=PS [Reflexive property] 

∠QPS=∠RSP [Right angles] 

So, △QPS≅ △RSP [Side-Angle-Side congruence criterion] 

Then PR=QS [Congruent parts of congruent triangles] 

So, the diagonals are congruent. 

Exercise

• Quadrilateral PQRS is a rectangle. Find the value of t.

• What is the perimeter of the parallelogram WXYZ?

• Give the condition required if the given figure is a rectangle.

• For rectangle GHJK, find the value of GJ.

• Find the angle perimeter of LOPNM.

Concept Map

What we have learned

• A parallelogram in which each pair of adjacent sides is perpendicular is called a rectangle.
• The diagonals of a rectangle are congruent.