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# 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,

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.

### 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.

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