QUADRILATERAL

QUADRILATERAL

In this chapter we will discuss on closed figure formed by four lines known as quadrilateral .A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) In an order.Since, ‘quad’ means ‘four’ and ‘lateral’ is for ‘sides’ therefore ‘quadrilateral’ means ‘a figure bounded by four sides’.Every quadrilateral has:

1. Four vertices,
2. Four sides
3. Four angles and
4. Two diagonals.

In the given figure

1. Four vertices are A,B,C,D.
2. Four sides are AB,BC,CD,DA.
3. Four angles are
4. Two diagonals are AC and BD.

A diagonals is a line segment obtained on joining the opposite vertices.

A closed plane figure formed by four line segments is called a quadrilateral.

AB, BC, CD and DA are four lines. They form a closed (area) figure. So ABCD is a quadrilateral and is being written as quad. ABCD or ABDC. The points A, B, C, D are called as its vertices. The four line segments AB, BC, CD, DA are the four sides and the four angles ∠A, ∠B, ∠C and ∠D are the angles of quadrilateral ABCD.

Two sides of a quadrilateral are consecutive or adjacent sides if they have a common point. That common point is one of the vertices of the quadrilateral. And two sides of a quadrilateral are said to be opposite sides if they have no common end point or vertex.

ANGLE SUM PROPERTY OF A QUADRILATERAL

THEOREM 1:

The sum of the four angles of a quadrilateral is 360°.

Given: A quadrilateral PQRS in which PR is its one diagonal.

To prove: ∠P + ∠Q + ∠R + ∠s = 360º

Proof : In ΔPQR,

⇒ ∠QPS + ∠QRS + ∠Q + ∠S = 360°

⇒ ∠P + ∠R + ∠Q + ∠S = 360°

i.e., ∠P + ∠Q + ∠R + ∠S = 360° Hence proved

question The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

Solution: Given the ratio between the angles of the quadrilateral = 3 : 5 : 9 : 13 and 3 + 5 + 9 + 13 = 30

Since, the sum of the angles of the quadrilateral = 360^o

First angle of it = 3/30× 360o = 36^o,

Second angle = 5/30× 360^o = 60^o,

Third angle = 9/30× 360^o = 108^o,

And,Fourth angle = 13/30× 360^o = 156^o

The angles of quadrilateral are 360^o, 60^o, 108^o and 156^o.

ALTERNATE SOLUTION:

Let the angles be 3x, 5x, 9x and 13.

3x + 5x + 9x + 13x = 360o

⇒ 30x = 360^o and x =360º/30 = 12^o

1st angle = 3x = 2 × 12^o = 360^o

2nd angle = 5x = × 12^o = 60^o

3rd angle = 9x = 9 × 12^o = 108^o

And,4th angle = 13 × 12^o = 156^o.

question Use the information given in adjoining figure to calculate the value of x.

Solution: Since, EAB is a straight line.

∠DAE + ∠DAB = 180^o

⇒ 73^o + ∠DAB = 180^o

i.e., ∠DAB = 180^o– 73^o = 107^o

Since, the sum of the angles of quadrilateral ABCD is 360^o

107^o + 105^o + x + 80^o = 360^o

⇒ 292^o + x = 360^o

⇒ x = 360^o– 292^o

⇒ x = 68^o Ans.