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.

 

Quadrilateral

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).
  •                   (AB, BC), (BC, CD), (CD, AD), (AD, AB) are four pairs of adjacent sides.
  •                   \[(\angle A,\,\,\angle B),\] \[(\angle B,\,\,\angle C),\] \[(\angle C,\,\,\angle D),\] \[(\angle D,\,\,\angle A)\] are four pairs of adjacent angles.
  •                  The sum of all angles of a quadrilateral is\[360{}^\circ \].

 

Types of Quadrilateral

  •                  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.

 

Other Topics

Notes - Geometry
  15 10


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