# 6th Class Mathematics Geometry

Geometry

Category : 6th Class

GEOMETRY

GEOMETRY

Geometry is derived from two greek words "Geo" means "Earth" metron means "Measurement". That means measurement of Earth is called geometry.

Basics terms of geometry

•                  There are three basics undefined terms of geometry.

(i) Point       (ii) Line       (iii) Plane

Point: Point is a mark of position, it is made by sharp tip of pen, pencil and nail.

•           It is denoted by capital letter.
•           It is represented by
•           A point has no length, no breadth and no thickness.

•                   Line segment: The distance between two points in a same plane is called a line segment.

•           It is denoted by$\overline{AB}$. It is measured in 'cm' or 'inch'.
•           It can be measured.
•          1 inch$=2.5\,\,cm,$

•                   Rays: A line segment extended endlessly in one direction is called a ray.

•            It is denoted by$\overline{OA}$.
•            It can't be measured.

•                   Line: A line segment extended endlessly in both directions is called a line.

•           It is denoted by$\overline{AB}$
•           It can't be measured.

•                   Plane: A smooth flat surface which extended endlessly in all the directions is called a "plane".

Example:

(i) Surface of a blackboard in your class room.

(ii) Floor of the classroom.

Note: A plane has length and breadth.

A plane has no thickens or boundary.

•                  Collmearity of points: Three points A, B, C in a plane are collinear if they lie on the same straight line.

•           $AB+BC=AC$

•                    Non-ColIinear points: The points which do not lie on the same line are called non - collinear point.

Note: Number of lines that can be drawn through 'n? non - collinear points is$\frac{n\left( n-1 \right)}{2}$.

Properties of lines

Passing through a point an infinite number of lines can be drawn.

${{l}_{1}},\,\,{{l}_{2}}$................... ${{l}_{n}}$ all pass through 'p'.

•                   These lines are called concurrent line and the point P is called the point of concurrence.
•                   Two lines in a plane are either intersecting or parallel.

ANGLE

•                     An angle is union of two different rays having the same initial point.

•                    Initial point is Q.
•                   Angle is denoted by$\angle PQR=30{}^\circ .$

•                     Unit of measurement of angle is degrees, which is represented as ${}^\circ$ (e.g., $60{}^\circ ,$ $70{}^\circ ,$ $80{}^\circ ,$etc.)

Types of Angle

•                Zero Angle: Initially the terminal ray coincides with initial ray without any rotation then the angle formed is a zero angle and its measure is$0{}^\circ .$

•                   Acute Angle: An angle is measure greater than $0{}^\circ$ but less than $90{}^\circ$ is called an Acute Angle.

•            Here, $\angle AOB=30{}^\circ$formed Acute Angle.
•                   Right Angle: An angle is measure $90{}^\circ$ is called a right angle.

•            Here, $\angle PQR=90{}^\circ ,$formed right angle.
•                  Obtuse Angle: An angle is measure greater than $90{}^\circ$ but less than $180{}^\circ$ is called an obtuse angle.

•            Here, $\angle PQR=120{}^\circ ,$formed obtuse angle.

•                    Straight Angle: An angle is equal to $180{}^\circ$is called a straight angle.

•          Here, $\angle PQR=180{}^\circ ,$formed straight angle.
•                   Reflex Angle: An angle is measure more than $180{}^\circ$ but less than $360{}^\circ$ is called a reflex angle.

•           Here, $\angle PQR=220{}^\circ ,$ formed reflex angle.
•                  Complete angle: An angle is equal to $360{}^\circ$is called a complete angle.

•            Here, $\angle AOB=360{}^\circ ,$formed complete angle.

Some more part of Angle

•                   Vertically opposite angle: It two lines are interested at a point then vertically opposite angles are equal.

•            $\angle POS=\angle QOR$and $\angle POR=\angle QOS$
•            $\angle 1=\angle 2$and$\angle 3=\angle 4$
•            $\angle POS=\angle QOR$ and $\angle POR=\angle QOS$ are formed vertically opposite angle.

•                    Adjacent angle: Two angles in a plane are said to be adjacent angles if they have a' common vertex.

•             In the adjoining figure x and y are called adjacent angles with common vertex A and common arm AC.

•                    Linear pair of angle: The pair of adjacent angles whose non common arms are two opposite rays is called a linear pair of angles.

•      In the adjoining figure $\angle PQS$ and $\angle RQS$ form a linear pair of angles.
•           $\angle 1+\angle 2=180{}^\circ$

•                    Complementary angle: if the sum of measures of any two angles is $90{}^\circ ,$then they are said to be complementary angles.

•           $\angle AOB+\angle PQR=30{}^\circ +60{}^\circ =90{}^\circ ~$

•                   Supplementary Angle: If the sum of any two angles is $180{}^\circ$then they are said to be supplement angles.

•            $\angle AOB+\angle PQR=60{}^\circ +120{}^\circ =180{}^\circ$

Parallel lines:

Two lines which never meet, even when they are extended infinitely, are known as parallel lines.

•             $l//m$
•             The distance between two parallel line be same.
•             The angle between two parallel line is$0{}^\circ$.

Transversal;

A straight line which intersect two or more given lines at different points is called a transversal.

Example:

Classification of angles formed by a transversal

If two parallel lines cut by a transversal then some angles are formed.

1.            Corresponding Angles.

$(\angle 2,\,\,\angle 6),\,\,(\angle 1,\,\,\angle 5),\,\,(\angle 3,\,\,\angle 7)$and$(\angle 4,\,\,\angle 8)$.

•      Pair of corresponding angles are equal.

2.            Alternate Interior Angles:

$(\angle 3,\,\,\angle 5);(\angle 4,\,\,\angle 6)$

•            Pair of alternate angles are equal.

3.            Alternate exterior angles:

$(\angle 1,\,\,\angle 7);(\angle 2,\,\,\angle 8)$

•           Pair of alternate exterior angles are equal.

4.            Co-interior angle or consecutive angle or allied angle.

$(\angle 4,\,\,\angle 5)$and$(\angle 3,\,\,\angle 6)$

•            Pair of co-interior angles are need not be equal.
•           Sum of co-interior angles is $180{}^\circ .$

$\angle 4+\angle 5=180{}^\circ ,$and$\angle 3+\angle 6=180{}^\circ ,$

Example: 1. In the given figure $AB||CD,$ $l$ is a transversal then$x=$?

Solution: $x=60{}^\circ$(by corresponding angle)

Example: 2. In the given figure $AB||CD||EF$ then $x+y$ is

Solution: $AB||CD$and BC be the transversal then $\angle ABC=\angle BCD$

$85{}^\circ =y{}^\circ +35{}^\circ$

$\therefore$$y=50{}^\circ$

Now, $EF||CD$and CE be the transversal then

$\angle DCE+\angle CEF=180{}^\circ$(by co-interior Angle)

$35{}^\circ +x=180{}^\circ$

$x=145{}^\circ$

$\therefore x+y=50+145{}^\circ =195{}^\circ$

Triangle

A closed figure bounded by three line segments is called a triangle.

•                   We read as 'triangle ABC" and it is denoted by$\Delta ABC.$
•                  Triangle have three sides and three angles. (These are called element of triangle)
•                  The sum of all angles of a $\Delta$ is $180{}^\circ$$\angle A+\angle B+\angle C=180{}^\circ$

Classification of triangles according to the side

1.            Equilateral triangle:

•            A triangle whose sides are equal in length is called an 'equilateral triangle'.
•           $AB=BC=AC$
•            All angle are equal.

$\angle A=\angle B=\angle C=60{}^\circ$

2.            Isosceles triangle:

•            A triangle in which two sides are equal in length is called an isosceles triangle.
•           $AB=BC$and $\angle B=\angle C$
•             BC is called base and $\angle B$ and $\angle C$ are called base angles.

3.            Scalene triangle:

•           If no two sides of a triangle are equal in length, it is called a scalene triangle.
•           $AB\ne BC\ne AC$

Classification of triangles according to the angles

1.            Acute angled triangle: If each angle of a triangle is an acute angle, men it is called an

Acute' angled triangle.

•            Measure of each angle is less than 90°.

2.            Right angled triangle:

A triangle in which one of its angles is a right angle is called a 'Right angled triangle'.

•           $\angle B=90{}^\circ$
•          The opposite side of the right angle is called Hypotenuse.
•          $A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}$(Pythagoras theorem)

3.            Obtuse angled triangle:

A triangle containing an obtuse angle is called an obtuse angled triangle.

•          $\angle B>90{}^\circ$

Median

A line segment in which joins a vertex of a triangle to the mid-point of the opposite side is called a median.

•             AD, BE and FC are called medians.
•             Point of intersection of the medians is called centroid. It is denoted by G.
•             'G' divide AD in the ratio $2:1$from the vertex side.

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

•                   AB, BC, CD and AD are four sides of ABCD quadrilateral.
•                   $\angle A,\,\,\angle B,\,\,\angle C$and $\angle D$ are four angles of the quadrilateral.
•                   The quadrilateral has two diagonals (AC and BD).
•                   $(\angle A,\,\,\angle B),$ $(\angle B,\,\,\angle C),$ $(\angle C,\,\,\angle D),$ $(\angle D,\,\,\angle A)$ are four pairs of adjacent angles.
•                  The sum of all angles of a quadrilateral is$360{}^\circ$.

•                  Trapezium: A quadrilateral having only one pair of parallel sides.

•             $AB||CD$

•                   Isosceles trapezium: A trapezium in which non - parallel sides are equal.

•           $AD=BC$
•           $AB||CD$

•                   Parallelogram: A quadrilateral having both pairs of opposite sides parallel and equal.

•           $AB||CD$and $AD||BC$
•           $AB=CD$and $AD=BC$
•           $\angle A=\angle C$and $\angle B=\angle D$

•                    Rhombus: A parallelogram in which all sides are equal.

•             $AB||CD$and $AD||BC$
•             $AB=BC=CD=AD$
•             Diagonals are perpendicular to each other.
•             $AO=CO$and $BO=DO$ (i.e. diagonals bisect each other)

•                     Rectangle: A parallelogram in which each angle is equal to$90{}^\circ .$

•            $AB||CD$and $AD||BC$
•           $AB=CD$and $AD=BC$
•            $\angle A=\angle B=\angle C=\angle D=90{}^\circ$
•            Diagonals are equal $AC=BD$
•             $~OA=OC=OD=OB$

•                    Square: A rectangle in which whose all sides are equal.

•            $AB||CD$and $AD||BC$
•            $AB=BC=CD=AD$
•            $\angle A=\angle B=\angle C=\angle D=90{}^\circ$
•             Diagonals bisect each other at$90{}^\circ$.
•            $AC=BD$

•                     Kite: A quadrilateral having two pairs of equal adjacent sides but unequal opposite sides is called a kite.

•             $AD=DB$and $AC=BC$
•             Diagonals are perpendicular to each other.
•             $\angle A=\angle B$
•             $OA=OB$

•                    Circle: A circle is a set of points in a plane whose distance from a fixed point is constant.

•           O is called center.
•           OA and OB are radius.
•           AB is called diameter.
•           Diameter $=2\times$ Radius
•           Diameter is the longest chord of circle.
•          CD is called tangent of circle.
•           PQ is called secant of a circle.

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##### Notes - Geometry

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