Quadrilaterals: class 9 math: 9 Math

Definition and types of Quadrilateral: 9th math

A closed polygon with four sides, four angles and four vertices is called A QUADRILATERAL.

Here, PS, PQ, QR and PS are four sides, P, Q, R and S are four vertices and ∠P, ∠Q, ∠R and ∠S are four angles formed at vertices of the given quadrilateral.

Lines joining opposite vertices of a quadrilateral are called DIAGONALS of quadrilateral

A quadrilateral can have maximum two diagonals.

Here diagonals are depicted in red and blue lines.

Types of Quadrilaterals

Types of Quadrilateral

Quadrilaterals can be divided into following types:

(1) Parallelogram

A quadrilateral in which opposite sides parallel and equal is called A Parallelogram

Rectangle

A quadrilateral with opposite sides equal and all angles equal is called a rectangle.

In other words a rectangle is a parallelogram in which every angle is a right angle.

In the given figure of rectangle

(i) Side AB = Side DC

(ii) And Side AD = side BC

(iii) Angle A = angle B = angle C = Angle D = 90⁰

(iv) Diagonal AC = Diagonal DB

(v) And, OA = OC = OD = OB

(vi) Angle AOB = angle DOC and Angle AOD = angle COB

Square

A rectangle with all sides equal and all angles equal is called A Square. The length of both of the diagonals of a square is also equal.

Kite

A type of quadrilateral in which exactly two pair of consecutive sides is equal in length is called a kite.

In the given figure of kite

(i) Side AB = side BC

(ii) And, side AD = side DC

(iii) Angle A = Angle C

(iv) But, angle B ≠ angle D

(v) Diagonal AC ⊥ BD

Properties of a kite

(a) Exactly two pair of consecutive sides is equal in length a kite.

(b) One of the diagonal bisects other in a kite.

(c) Diagonals are perpendicular to one another in a kite.

(d) Angles are equal where two pair of sides meets in a kite.

Rhombus

A parallelogram with all the four sides are equal in length is called a rhombus.

In other words, a quadrilateral in which opposite sides parallel, all sides are of equal length and opposite angles are equal in measure is called a rhombus.

In other words rhombus is parallelogram in which all sides are equal and opposite angles are equal.

In the given figure of Rhombus

(i) AB || DC and AD || BC

(ii) And AB = BC = CD = AD

(iii) Angle A = angle C and Angle D = angle B

(iv) AO = OC and OD = OB

Trapezium

A quadrilateral in which a pair of sides is parallel is called trapezium.

In the given figure of trapezoid

(i) Side AB || DC

(ii) DE is the height of the trapezium

(iii) When two sides of a trapezium are equal, it is called Isosceles trapezium.

(iv) The red arrow shows the parallel sides.

Angle Sum Property of Quadrilateral

Theorem 8.1 (Class nine math Quadrilateral)

The sum of the angles of a quadrilateral is 360^o.

Given,

Let PQRS is the given quadrilateral

Then to prove

∠P + ∠Q + ∠R + ∠S = 360^o

Construction

Points P and R has been joined.

Since, P and R are opposite vertices of a quadrilateral, thus PR is a diagonal.

Proof

Now, in triangle PRS;

We know from angle sum property of a triangle which says that sum of angles of a triangle = 180^o.

Thus, ∠SPR + ∠PRS + ∠RSP = 180^o - - - - (i)

Now, in triangle PQR,

∠PQR + ∠QRS + ∠QRS = 180^o - - - - - (ii)

[Because sum of angles of a triangle = 180^o]

Now, from equation (i) and equation (ii), we get

∠SPR + ∠PRS + ∠RSP + ∠PQR + ∠QRS + ∠QRS = 180^o + 180^o

⇒ ∠SPR + ∠RPQ + ∠PRS + ∠QRS + ∠RSP + ∠PQR = 360^o

⇒ ∠P + ∠R + ∠S + ∠Q = 360^o

[Because, ∠SPR + ∠RPQ = ∠SPQ = ∠P

And, ∠PRs + ∠PRS = ∠QRS = ∠R

And, ∠RSP = ∠S

And, ∠PQR = ∠Q]

⇒ ∠P + ∠Q + ∠R + ∠S = 360^o Proved

Thus, the sum of the angles of a quadrilateral is equal to 360^o Proved

Theorem 8.2 (Class nine math Quadrilateral)

A diagonal of a parallelogram divided it into two congruent triangles.

As given, Let PQRS is a parallelogram.

And PQ is the diagonal of this given parallelogram.

This diagonal divides the parallelogram in two triangles.

Then, to prove that Triangle PQR and Triangle PRS are congruent.

Proof

Since, PQRS is a parallelogram

Thus, SR || PQ and PS ||QR

[Since opposite sides of a parallelogram are parallel.]

Now, in ΔPRS and ΔPQR

PS||QR

And PRs is a transversal which intersects these two parallel sides or line at point P and R respectively.

Now, we know that "if a transversal intersects two parallel lines, then each pair of alternate interior angles are equal."

Here, ∠SPR and ∠QRP are a pair of alternate interior angles, hence are equal.

i.e. ∠SPR = ∠QRP

Similarly, ∠RPQ and ∠PRS are a pair of alternate interior angles

Thus, ∠RPQ = ∠PRS

And, side PR is common in both of the triangles

Thus, here two angles and side between them are equal in both of the triangles.

Thus, according to Angle Side Angle (ASA) congruency

ΔPRS ≅ ΔPQR Proved

Therefore, A diagonal of a parallelogram divides it into two congruent triangles Proved

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