6th Class Mathematics Practical Geometry Geometry

Geometry

Category : 6th Class

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\[360{}^\circ \].