Geometry

 Mind Maps

Class V - maths: Geometry
Q) What is a collection of points extended endlessely in both the directions along a straight path?

Q) Which lines pass through the same point?

Q) What we call the points, if three or more points lie in a straight line?

Q) What we call if three points not lying in the same straight line?

Q) What is a flat surface that goes on endlessly or extends indefinitely in all directions?

Q) A triangle with two sides equal is called an?

Q) Geometry is a branch of mathematics that deals with?

Q) The portion of the circle divides it into two equal arcs, each of the arc is called a?

Q) The below image shows the?

Q) The below image shows the?

Q) The distance between the center and any point on the?

Q) What is a protractor?

Q) What is Acute angle and draw the diagram?

Q) What is Reflex angle and draw the diagram?

Q) What is geometry?

Q) What is circle?

Q) What is Polygon?

Q) A part of a line that has 2 end points is a?

Q) The sum of the angles of a triangle is equal to ?

Q) Explain briefly about the types of lines?

Q) Explain briefly about the Classification of Triangles?

Q) Explain briefly about the classification of angles?

Q) Explain about Protractor and its use? Write the steps to measure the angle?

Q) Explain about the properties of Triangle? What is Polygon?

Q) What is reflection between point and lines? Give an examples?

Q) What is circle and give an example?

Q) What is Polygon and give an example?

GEOMETRY

Definition:

A branch of mathematics that deals with points, lines, angles, surfaces, and solids

Example:

Let us study the geometrical shapes in detail.

POINT AND LINES

Point and line are the building blocks of geometry. These two concepts form the basis of geometrical concepts and theories.

Let us begin with a point.

Point: ∠ = 23⃗

$\stackrel{\to }{\mathrm{23}}$
• In geometry, a point is an undefined concept.
• A mathematical point is a fixed location. It cannot be moved.
• A point may be represented by drawing a dot on the paper.
• The dot is not a mathematical point, it only suggests a point.

Example:

The tip of a needle or a drawing pin can be considered as a physical picture of a point

Capital letters of the English alphabet are used to show different points represented by dots

A single capital letter is used to represent a point.
Space may be considered as the set of all points or locations

Line:

A line is a collection of points extended endlessly in both the directions along a straight path.

A line has no beginning and no end. It has no end points.

• When the word line is used, it means straight line.
• A line is represented as a double arrow showing that it can be extended without limit in both directions.
• "↔"is the symbol of a straight line.
• To name a line, any two points on the line are selected and named by capital letters of the English alphabet.

• The line above is written as  bar (XY)

Line Segment

• A line segment is a part of a line.
• It has two end points.
• " "is the symbol of a line segment.
• Line segments are named by putting a bar over the letters of the two end points.
• The segment shown is named:

• The length between the two points is limited. It cannot be increased or decreased.

Ray

• A ray is a part of a line.
• It has one end point and goes endlessly in one direction.
• It is represented by an arrowhead on one side and a point on the other.
• To name a ray, the end point is named first.
• The ray shown above can be read as CD.
• Each line contains an unlimited number of rays like  bar (CD), CB, DE etc.

RELATION BETWEEN POINT AND LINES

Infinite number of lines can be drawn passing through a point.

"Lines that pass through the same point are known as concurrent lines".

Example:

Collinear Points

"If three or more points lie in a straight line, the points are said to be collinear points"

Example:

• Here, A, B, and C are points on AC.
• Point B is in between points A and C.
• So, A, B, and C are collinear points.

Non-Collinear Points

Three points not lying in the same straight line are called the non-collinear points"

Example:

Here, the points A, B,C and P, Q, R are non-collinear points.

"A plane is a flat surface that goes on endlessly or extends indefinitely in all directions".

Example:

• A plane is a collection of an infinite number of points.
• A plane contains an unlimited number of lines, line segments and rays
ANGLE

"Two rays with a common end point form an angle"

Example:

• Point B is the common end point of BA and BC.
• The figure we get is angle ABC.
• The common end point B is called the 'Vertex of the angle'.
• AB and BC are the arms of the angle.
• The symbol of angle is"∠"

Naming An Angle

• An angle can be named in different ways. We can name an angle by using three letters of the English alphabet
• Example:

• When three letters are used to name an angle, the letter showing the vertex is always written in the middle.
• An angle can also be named by the vertex
• Example:

∠B,∠Q

Example:

∠2,etc∠

• Angles can also be named by assigning a numeral to the angle

Look at the following angle

In the above figure, the angles formed are:

Example:

∠ABD,etc∠DBC,∠ABC

INTERIOR AND EXTERIOR OF AN ANGLE

An angle has two parts:

1. Interior part
2. Exterior part.
• The points lying inside the angle form the interior of the angle.
• The points lying outside the angle form the exterior of the angle
• Example:

PROTRACTOR

"A protractor is an instrument which is used to measure an angle"

• It has two scales of measurement, the inner scale and the outer scale.
• Angles from 0° to 180° are marked on both the scales.
• One scale is read clockwise, and the other is read anticlockwise.

Example:

Measurement Of Angles

Degree

It is the unit of measurement in measuring angles.

The symbol of degree is " ° ".
We write angles as 10°, 40°, 80° etc.
The number of degrees in an angle is called its measure

Example:

In the above example ∠FOE is 20

So, the measure of,∠FOE = 20°

The notation for measuring an angle in degrees, such as angle FOE is;

'm∠FOE = n', where 'n' is a number representing the measure of the angle.

The symbol 'm∠FOE' is read as the 'measure of angle FOE'.

To measure an angle, following steps are to be followed:

1. Place the centre point of the protractor on the vertex Q of the angle.
2. Adjust the protractor in such a way that the base line coincides with the arms of the angle.
3. Check the scale where the baseline arm points to 0°.
4. Read the measure of the angle where the other arm crosses the scale.

Example:

Here,the measure of∠ABC = 40°.

CLASSIFICATION OF ANGLES

Angles are classified on the basis of their measure

CONSTRUCTING ANGLES WITH A PROTRACTER

To construct an angle using a protractor, following steps are to be followed:

Example

Construct∠ABC = 50°.

Steps:

1. Draw a ray →BC
2. Place the centre of the protractor on the end point B of BC. Point B will form the vertex point.
3. Adjust the protractor such that the base line is along BC.
4. Now measure carefully from the 0° to the opposite of the vertex on the inner scale.
5. Mark point A at 50°
6. Remove the protractor and join points A to B, to get BA.
7. The measure of ∠ABC =50°.

Example

• onstruct∠PQR = 120°.
• Draw a ray QP.
• Place the protractor carefully on QP such that the centre of the protractor lies on the point of vertex Q.
• Adjust the protractor such that the base line is long QP.
• Measure carefully from 0° on the outer scale.
• Mark point R at 120°.
• Remove the protractor and join R to Q to get QR.
• m∠PQR = 120

Some more examples of various angles are:

TYPES OF LINES

The types of lines are as follows:

1. Coplanar lines
2. Parallel lines
3. Intersecting lines
4. Perpendicular lines
5. Let us discuss them in detail

Coplanar Lines

Two lines lying in the same plane are called "coplanar lines"

MN and PQ are coplanar lines.

Parallel Lines

• "Two or more coplanar lines that do not intersect are called parallel lines".
• Parallel lines are equidistant from each other
• Symbol for parallel lines are"||"

Example

Railway tracks, banks of a canal, opposite edges of a book are some examples of parallel lines from day - to - day life.

Parts of parallel lines are also parallel, i.e.,Line segments or rays can also be parallel.

Parallel lines can be vertical,       horizontal or oblique

Intersecting Lines

"Two or more lines that cross each other are called intersecting lines".

Two intersecting lines have only one point in common.

AE and BD intersect at point C.

Two lines can intersect on only one point

Example

The point where the lines intersect is known as the point of intersection.

Perpendicular Lines

Two lines that intersect or meet each other to form a right angle are called" perpendicular lines"

Example

In the first figure, CD⊥AB.

In the second figure, AB⊥MN.

POLYGON

definition

"A closed figure which is made up of three or more line segments that do not cross each other is called a polygon".

Example

• A triangle is formed by joining three non-collinear points.
• So, a triangle is a polygon made up of 3 line segments
• A square and a rectangle are examples of a quadrilateral.
• A pentagon is a polygon made up of five line segments.
TRIANGLE

Definition

A triangle is a polygon made up of three sides. This is the least number of sides needed to form a triangle".

• It has three vertices and three sides.
• An angle is named by the vertex point in the middle.
• The total measure of all the three angles of a triangle is equal to 180o.
• The symbol of a triangle is"∆"

Example

Classification Of Triangles

Triangles can be grouped together and named according to their sides.

Let us have a look at different triangles grouped together according to their lengths.

Triangles can also be grouped together according to the measure of their angles.

The total measure of the angles in a triangle is equal to 180°

PROPERTIES OF TRIANGLE
• The sum of angles of a triangle is equal to 180°
• Example

Here, 70°+ 70°+ 40° = 180°.

• The sum of the measure of any two sides of a triangle is always greater than the third side

Example

Here, 5cm + 5 cm = 10 cm > 4cm.
5 cm + 4 cm = 9 cm > 5 cm

CIRCLES

A circle is a closed figure made upof curved lines

Example

Parts Of A Circle

Let us discuss the various parts of a circle.

1. Circumference:

2. It is the distance around the circle.

4. The distance between the centre and any point on the circumference.

5. Diameter:

6. The distance from a point of the circumference to another point of the circumference passing through the centre. It is double the radius.

7. Chord:

8. The distance between one point of the circumference to another point of the circumference. It may or may not pass through the centre.

9. Arc:

10. A part of the circumference of a circle is called an arc. It is named by three points.

11. Interior And Exterior Of A Circle:

12. The area enclosed by the circumference is called the interior of the circle. The area outside the circumference is called the exterior of the circle.

13. Semicircle:

14. The portion of the circle divides it into two equal arcs, each of the arc is called a semicircle.

Drawing A Circle Of Given Radius:

• Draw a circle of radius 4 cm.

Steps:

• Take a sharp pencil and a compass.
• Fix the pencil in the compass.
• Stretch the arms of the compass and adjust the opening between the pointed ends
• Keep the metal point fixed on the paper and move the pencil arm around it as shown in the figure below.
• The figure thus constructed is a circle of radius 4 cm
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