Quadrilateral

What is a Quadrilateral

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What is a Quadrilateral

A closed figure bounded by four line segments is called a quadrilateral.
quadrilateralAll closed figures shown in figure are quadrilaterals as all of them have four line segments. A quadrilateral has four sides, four vertices, and four angles. Let us consider a quadrilateral PQRS.
quadrilateral-1

  1. Sides of the quadrilateral PQRS are PQ, QR, RS, SP, or QP, PS, SR, RQ.
  2. Angles of the quadrilateral PQRS are ∠P, ∠Q, ∠R, ∠S or ∠SPQ, ∠PQR, ∠QRS, ∠RSP.
  3. Vertices of quadrilateral PQRS are P, Q, R, and S.

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In addition to the above mentioned features, a quadrilateral also has two diagonals. The line segments joining the opposite vertices of a quadrilateral are called its diagonals. So, PR and QS are diagonals of quadrilateral PQRS that divide the quadrilateral into four triangles.
quadrilateral-2

Adjacent sides
In quadrilateral PQRS, the two sides PQ and QR having a common end point Q, are called adjacent sides. Similarly, QR and RS, RS and SP, SP and PQ are adjacent to each other.
quadrilateral-3 Opposite sides
The sides PQ and RS, PS and QR are opposite to each other. They are the opposite sides.
They do not have any common point.

Adjacent angles
Two angles of a quadrilateral that have a common arm are called adjacent angles.
In quadrilateral PQRS
∠P and ∠S, ∠P and ∠Q, ∠Q and ∠R, ∠R and ∠S are pairs of adjacent angles.

Opposite angles
Two angles of a quadrilateral that have no common arm are called opposite angles.
In quadrilateral PQRS, ∠P and ∠R, and ∠Q and ∠S are pairs of opposite angles.

Types of Quadrilaterals:

Quadrilaterals are classified as the following, depending upon their sides.

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

https://www.youtube.com/watch?v=dClELYwm8Nc

Interior and exterior of a quadrilateral

A quadrilateral divides a plane into three parts:

  1. Interior region: The region inside the quadrilateral is called the interior of the quadrilateral.
    Interior-exterior-quadrilateral Points A, B, and C are in the interior of the quadrilateral PQRS.
  2. On the quadrilateral: The points M and N which lie on the boundary are said to be on the quadrilateral.
  3. Exterior region: The region outside the quadrilateral is called the exterior of the quadrilateral. Points U and T are in the exterior of the quadrilaterals.

 

Quadrilateral Family

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Quadrilateral Family

Each member of the quadrilateral family will describe its specific properties.

Quadrilateral

  • I have exactly four sides.
  • The sum of the interior angles of all quadrilaterals is 360º.
    Quadrilateral Family 1
  • A quadrilateral is any four sided figure. Do not assume any additional properties for a quadrilateral unless you are given additional information.

Trapezoid

  • I have only one set of parallel sides.
    [The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.]
    Quadrilateral Family 2
  • A trapezoid has ONLY ONE set of parallel sides. When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are not parallel.

Isosceles Trapezoid

  • I have: only one set of parallel sides
  • base angles congruent
  • legs congruent
  • diagonals congruent
  • opposite angles supplementary
    Quadrilateral Family 3
  • Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

Parallelogram

  • I have: 2 sets of parallel sides
  • 2 sets of congruent sides
  • opposite angles congruent
  • consecutive angles supplementary
  • diagonals bisect each other
  • diagonals form 2 congruent triangles
    Quadrilateral Family 4
  • Notice how the properties of a parallelogram come in sets of twos: two properties about the sides; two properties about the angles; two properties about the diagonals. Use this fact to help you remember the properties.

Rectangle

  • I have all of the properties of the parallelogram PLUS
  • 4 right angles
  • diagonals congruent
    Quadrilateral Family 5
  • If you know the properties of a parallelogram, you only need to add 2 additional properties to describe a rectangle.

Rhombus

  • I have all of the properties of the parallelogram PLUS
  • 4 congruent sides
  • diagonals bisect angles
  • diagonals perpendicular
    Quadrilateral Family 6
  • A rhombus is a slanted square. It has all of the properties of a parallelogram plus three additional properties.

Square

  • I have all of the properties of the parallelogram AND the rectangle AND the rhombus.
    Quadrilateral Family 7
  • The square is the most specific member of the quadrilateral family. It has the largest number of properties.

Convex and Concave Quadrilaterals

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Convex and Concave Quadrilaterals

Convex quadrilateral: A quadrilateral is called a convex quadrilateral, if the line segment joining any two vertices of the quadrilateral is in the same region.
In figure, ABCD is a convex quadrilateral because AB, BC, CD, DA, AC, BD are in the same region of the quadrilateral.
Convex-QuadrilateralIn a convex quadrilateral each angle measures less than 180°.

Concave quadrilateral: A quadrilateral is called a concave quadrilateral, if at least one line segment joining the vertices is not a part of the same region of the quadrilateral.
Concave-QuadrilateralThat is, any line segment that joins two interior points goes outside the figure. In a concave quadrilateral at least one angle is a reflex angle, i.e., an angle larger than 180°. In figure, ABCD is a concave quadrilateral because a line joining the vertices A and C is going outside the quadrilateral region.

Angle sum property of a quadrilateral

The sum of all angles of a quadrilateral is 360° or four right angles.
Draw a quadrilateral ABCD with one of its diagonals AC.
Angle-sum-property-quadrilateral
Diagonal AC divides the quadrilateral into two triangles, i.e., ΔADC and ΔABC.
Clearly ∠l + ∠2 = ∠A
and ∠3 + ∠4 = ∠C            …(1)
We know that the sum of the angles of a triangle is 180°.
∴ In ΔABC, ∠1 + ∠3 + ∠B = 180°
In ΔADC, ∠2 + ∠4 + ∠D = 180°
Sum of the angles of a quadrilateral
= Sum of the angles of ΔABC and ΔADC
∴ (∠1 + ∠3 + ∠B) + (∠2 + ∠4 + ∠D) = 180° + 180°
or ∠1 + ∠3 + ∠B + ∠2 + ∠4 + ∠D = 360°
or (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360°
or ∠A + ∠B + ∠C + ∠D = 360°         (using 1)
Hence, the sum of the angles of a quadrilateral equals 360°.

Example 1: The angles of a quadrilateral are in the ratio of 1 : 2 : 1 : 2. Find the measure of each angle.
Solution: Let the first angle of a quadrilateral be x.
Here, second angle = 2x
third angle = x
fourth angle = 2x
Sum of all angles of a quadrilateral = 360°
∴ x + 2x + x + 2x = 360°
6x = 360°
x = 60°
∴ First angle = x = 60°
Second angle = 2x = 2 × 60° = 120°
Third angle = x = 60°
Fourth angle = 2x = 120°.