Mathematics

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 1 Introduction 2 What is a quadrilateral? 3 Types and Properties of Quadrilaterals 5 Theorems relating to Quadrilaterals 6 Example Problems 7 Summary 8 FAQs

29 January 2021

“Geometry is an art of correct reasoning from incorrectly drawn figures.”

-Henri Poincare

The word “Quadrilateral” is derived from the Latin language, in which “Quadri” means a variant of four and “Latus” means “side”.

A quadrilateral is a polygon with four edges (sides) and four vertices (corners). Let us understand the properties and applications of quadrilaterals. Here is a downloadable PDF to explore more.

A Quadrilateral is a flat-surfaced polygon with four sides and four corners. The interior angles of the Quadrilateral always add up to 360°.

Quadrilaterals are categorized into 6 types depending on their sides. They are:

1. Parallelogram
2. Trapezium
3. Rhombus
4. Kite
5. Rectangle
6. Square

Different Types and Properties of Quadrilateral

1. Parallelogram

A Quadrilateral in which opposite sides are parallel and equal is called a Parallelogram.

Properties of Parallelogram

• In a Parallelogram,
1.  Opposite sides are equal.
2.  Opposite angles are equal.
3.  Each diagonal bisects the parallelogram.
4. The opposite sides and angles are congruent.
• The consecutive angles of the parallelogram are supplementary i.e., they add up to 1800.
• The diagonals bisect each other.
• Area of the Parallelogram = Base$$\times$$Height

Where Base is any side of the Parallelogram.

Height is the perpendicular length from the opposite vertex to the base.

1. Trapezium

A Quadrilateral in which one pair of opposite sides are parallel but the other two sides of it are non-parallel is called Trapezium. A trapezium is also called Trapezoid.

Properties of Trapezium

• The two angles of the adjacent sides of the Trapezium are supplementary.
• The diagonals of the Trapezium do not bisect each other at right angles.
• If the non-parallel sides of the Trapezium are equal, it is called an Isosceles Trapezium.

• In Isosceles Trapezium,
1. The adjacent angles of the parallel sides are equal.
2. The diagonals are equal.
• Area of the Trapezium $$=\frac{1}{2}\times$$ Sum of the parallel sides $$\times$$ Height

Where Height is the perpendicular distance between two parallel sides.

1. Rhombus

The quadrilateral in which all the sides are equal is called a Rhombus.

Properties of Rhombus

• The diagonals of the rhombus bisect each other at 900.
• Any pair of adjacent sides are equal.
• The opposite sides of the Rhombus are parallel to each other.
• The opposite angles of the Rhombus are equal to each other.
• The sum of the two adjacent angles is equal to 1800.
• Area of the Rhombus $$= \frac{(d_1 \times d_2)}{2}.$$

Where, $$d_1$$ and $$d_2$$ are the diagonals of the Rhombus.

1. Rectangle

A quadrilateral in which the opposite sides are equal is called a Rectangle.

Properties of Rectangle

• Each interior angle of the Rectangle is 900.
• The diagonals of the rectangle are equal.
• The diagonals bisect each other.
• The opposite sides of the Rectangle are parallel to each other.
• Area of the Rectangle $$=$$ Length $$\times$$ Breadth.
1. Square

A quadrilateral in which all the sides and angles are equal is called a Square.

Properties of Square:

• All the sides of the Square are congruent to each other.
• The opposite sides of the Square are parallel to each other.
• The interior angle of the Square is 900.
• The diagonals of the square are equal.
• The diagonals bisect each other at 900.
• The diagonal cut the Square into two equal isosceles triangles.
• The length of the diagonal is greater than the length of its side.
• Area $$= a \times a$$

Where a is represented as the side of the Square.

1. Kite

A quadrilateral in which the two pairs of disjoint adjacent sides are equal is called a Kite.

Properties of Kite

• The kite is symmetrical about its long diagonal.
• Angles between the unequal sides of the Kite is equal.
• Diagonals of the Kite intersect each other at 900.
• The longer diagonal of the kite is the perpendicular bisector of the shorter diagonal of the kite.
• The shorter diagonal in the kite divides the kite into two isosceles triangles.
• Area of the Kite $$= \frac{1}{2}\times (D_1 \times D_2)$$

Where, $$D_1$$ is the longer diagonal of the Kite.

$$D_2$$ is the shorter diagonal of the Kite.

1. In a parallelogram,
1. Opposite sides are equal.
2.  Opposite angles are equal.
3.  Each diagonal bisects the parallelogram.
2. The diagonal of the parallelogram bisect each other.
3. The diagonals of the rectangle are equal and bisect each other.
4. The diagonals of the rhombus bisect each other at 900.
5. The diagonals of the square bisect each other at 900.

Example Problems

 Example 1

If the angles of the quadrilateral are X, X+20, X-40 and 2X. What is the difference between the greatest angle and smallest angle?

(a) $$70^\circ$$

(b) $$90^\circ$$

(c) $$80^\circ$$

(d) None of these

Answer: All the interior angles of the quadrilateral $$= 360^o$$

\begin{align}X+X+20+X-40+2 x&=360^{\circ} \6pt]5 X-20&=360^{\circ} \\[6pt]5 &X=360^{0}+20^{\circ} \\[6pt]5 X&=380^{\circ} \\[6pt]X&=76^{0}\end{align} Therefore, the answer is Option (d)  Example 2 The diagonals of a rectangle PQRS intersect at O. If $$∠ROQ=60^\circ$$, then find ∠OSP. (a) $$70^\circ$$ (b) $$50^\circ$$ (c) $$60^\circ$$ (d) $$80^\circ$$ Answer: $$∠ROQ=60^\circ$$ $$∠POS = 60^\circ$$ (Vertically Opposite Angles) In Triangle POS, OP=OS (Diagonals of the Rectangle bisect each other) Therefore, $$∠OSP = ∠SPO$$ So, \[\begin{align}\angle OSP+\angle SPO+\angle POS&=180^{\circ} \qquad\left(\text { Sum of Interior Angles of triangle is } 180^{\circ}\right)\\[6pt]\angle OSP+\angle OSP+\angle POS&=180^{\circ}\\[6pt] 2 \angle OSP&=180^{\circ}- \angle POS\\[6pt]2 \angle OSP=180^{0}-60^{0}&=120^{\circ}\\[6pt]\angle OSP&=60^{\circ}\end{align}

Therefore, the answer is option (c).

 Example 3

The bisectors of the angle of the parallelogram form ___________________

(a) Trapezium

(b) Rectangle

(c) Rhombus

(d) Kite

 Example 4

The measure of all the angles of the parallelogram, if an angle adjacent to the smallest angle is 240 less than twice the smallest angle is.

(a) $$37^{\circ}, 143^{\circ}, 37^{\circ},143^{\circ}$$

(b) $$108^{\circ}, 72^{\circ}, 108^{\circ},72^{\circ}$$

(c) $$68^{\circ}, 112^{\circ}, 68^{\circ},112^{\circ}$$

(d) None of these

Answer: Let the smallest angle of the parallelogram be $$x^\circ$$.

The adjacent angle is $$2x^{\circ}-24^{\circ}$$

The sum of the adjacent angles of the parallelogram is $$180^{\circ}.$$

What is the sum of all the interior angles of the quadrilateral?

Sum of all the interior angles of the quadrilateral $$= 360^\circ$$

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