Notes: Geometry

Table of Contents
1. Basic Geometrical Terms
2. Types of Lines
3. Measuring a Line Segment
4. Angle
5. Measure of an Angle
6. Measuring an Angle (Using a Protractor)
7. Drawing Angles using a Protractor
8. Types of Angles
9. Equal or Congruent Angles
10. Open and Closed Shapes
11. Tangrams
12. Triangles
13. Area and Perimeter of a Triangle
14. Right-Angled Triangles and the Pythagoras Theorem
15. Circle
16. Summary
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Introduction

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. It is fundamental to art, architecture, engineering and many branches of science. A clear understanding of basic geometric ideas is essential for teaching mathematics at the primary and upper-primary levels.

Basic Geometrical Terms

Point

A point denotes an exact location in a plane or in space. It has no length or breadth and is represented by a single capital letter, for example, A, B, C.

Line segment

Mark two points on a sheet and join them with a ruler. The portion of the straight line between these two points is a line segment. It has two endpoints and is named by its endpoints, for example, line segment AB.

Line

A line is a straight path that extends infinitely in both directions. It has no endpoints and is named by any two distinct points on it, for example, line AB (often written as AB with a double-headed arrow above when using notation). A straight line can be vertical, horizontal or slanting.

Ray

A ray starts at a point (its endpoint) and extends endlessly in one direction. The notation ray AB means the ray begins at A and passes through B. The arrow → in diagrams indicates the ray extends without end in that direction.

Plane

A plane is a flat surface that extends without end in all directions. We usually work with a finite part of a plane; points and lines lie on a plane. A plane can be named by any three non-collinear points on it.

Some real-life representations of a plane surface:

Types of Lines

• Parallel lines: Two lines in the same plane that never meet, however far they are extended, are parallel. They remain the same distance apart. The symbol "||" denotes "is parallel to".

Example notation: line XY || line PQ.

Everyday examples of parallel lines:

• Intersecting lines: Lines that cross each other at a point are intersecting lines. The point where they meet is called the point of intersection.

Examples:

• Perpendicular lines: When two intersecting lines meet to form right angles, they are perpendicular. The right angle is usually indicated by a small square at the intersection. The symbol '⊥' denotes "is perpendicular to".

Example notation: AB ⊥ CD.

Measuring a Line Segment

To measure the length of a line segment using a ruler

Let us measure line segment AB.

Step 1: Place the ruler along the segment so that the zero mark coincides with point A.

Step 2: Read the mark at point B on the ruler.

Example: If B lies at the 5.5 cm mark, then the length AB = 5.5 cm.

To draw a line segment of given length (for example, 6.8 cm)

Step 1: Mark a point A on paper.

Step 2: Place the ruler with its 0 mark at A.

Step 3: From A, measure along the ruler to the mark for 6.8 cm and mark point B. Join A and B to get segment AB of length 6.8 cm.

Angle

An angle is formed by two rays that meet at a common endpoint. The common endpoint is called the vertex and each ray is called an arm of the angle.

Some pictures help form an idea of an angle:

The symbol for angle is ∠.

Naming an angle

• Using three letters: a point on one arm, the vertex, and a point on the other arm, for example ∠PQR or ∠RQP.
• Using one letter: just the vertex, for example ∠Q (useful when only one angle has that vertex).
• Using a number assigned to the angle, for example ∠1.

Interior and exterior of an angle

The interior of an angle is the region between its arms. The exterior is the region outside the arms. Points lying on the arms are neither interior nor exterior.

Measure of an Angle

By the measure of an angle we mean the amount of rotation or turning from one arm to the other.

Angles are measured in degrees (°). One complete rotation measures 360°.

The standard unit for angles is the degree. The symbol for degree is "°".

Edurev Tips: One complete rotation is divided into 360 equal parts. Each part measures 1°.

Clock hands: The turning of a clock hand illustrates rotation and angle measure. One complete rotation of a hand is 360°.

Example 1

If a circle is divided into 20 equal parts, what is the measure of each part?

Angle formed by one complete rotation = 360°.

Measure of angle of each of the 20 equal parts = 360° ÷ 20 = 18°.

Example 2

Calculate the angle between neighbouring spokes on each wheel:

(a) Wheel divided into 4 equal parts: angle = 360° ÷ 4 = 90°.

(b) Wheel divided into 3 equal parts: angle = 360° ÷ 3 = 120°.

(c) Wheel divided into 8 equal parts: angle = 360° ÷ 8 = 45°.

Measuring an Angle (Using a Protractor)

The instrument used for measuring angles is called a protractor.

A protractor is usually a semicircle divided into 180°. The inner scale goes from 0° to 180° anticlockwise and the outer scale goes from 0° to 180° clockwise. Each small division equals 1°.

Edurev Tips: Choose the scale (inner or outer) so that one arm aligns with 0°; then read the other arm on the same scale.

To measure an angle less than 180°

Step 1: Place the centre (midpoint) of the protractor at the vertex of the angle.

Step 2: Make sure the 0° line of the protractor lies along one arm of the angle.

Step 3: Read the degree measure indicated by the other arm on the same scale. For example, if the other arm points to 60° then ∠BAC = 60°.

To measure an angle greater than 180° (a reflex angle)

If you are required to measure a reflex angle ∠BOC (greater than 180°):

Step 1: Measure the smaller adjacent angle a (less than 180°).

Step 2: Reflex angle = 360° - a°.

Drawing Angles using a Protractor

To draw an angle of 80°:

Step 1: Draw any ray OA.

Step 2: Place the protractor so that its centre is at O and align OA with 0° on one of the scales.

Step 3: Mark the point on the paper corresponding to 80° on the same scale. Label that point B.

Step 4: Remove the protractor and join B to O. Then ∠AOB = 80°.

When using the outer scale instead of the inner scale, the numeric markings you read change but the geometric angle remains the same; choose the scale consistently depending on ray alignment.

Types of Angles

Angles are classified by their measures:

• Acute angle: measure less than 90°.
• Right angle: measure exactly 90°.
• Obtuse angle: measure greater than 90° but less than 180°.
• Straight angle: measure exactly 180°.
• Reflex angle: measure greater than 180° but less than 360°.
• Full rotation: measure exactly 360°.

Equal or Congruent Angles

When two angles have the same measure, they are called equal or congruent. We write ∠ABC = ∠DEF to indicate the two angles are equal.

Open and Closed Shapes

Open shapes are those that do not begin and end at the same point.

Closed shapes begin and end at the same point. A closed shape is called simple if it does not cross itself.

Examples of open shapes:

Examples of closed shapes:

Simple closed figures you can draw without lifting the pencil and without crossing: A, B, D, G and H in the referenced figure - these are simple closed figures, while others that cross themselves are not simple.

Tangrams

A tangram is a Chinese dissection puzzle formed from seven plane shapes called tans: five triangles of various sizes, one square and one parallelogram. These seven pieces can be arranged to make many different shapes and figures; collectively they can form a square.

Tangram

Why use tangrams?

Tangrams are useful teaching tools because they:

• develop spatial reasoning and visualisation skills,
• teach shape recognition and symmetry,
• help understand area and perimeter informally,
• foster problem-solving and creative thinking.

Research and classroom experience show that working with tangrams can improve children's geometrical thinking and problem solving. There are more than 6,500 distinct shapes and pictures that can be formed using the seven tans.

Triangles

A triangle is a simple closed figure formed by three straight line segments joining three non-collinear points.

Every triangle has:

• Three vertices - the corner points where sides meet (for example A, B, C in ∆ABC).
• Three sides - the line segments AB, BC and CA.
• Three angles - the interior angles at the three vertices (for example ∠BAC, ∠ABC, ∠BCA).

Classification of triangles

Triangles can be classified in two main ways:

• Based on sides: scalene (all sides different), isosceles (two sides equal), equilateral (all three sides equal).
• Based on angles: acute-angled (all angles < 90°),="" obtuse-angled="" (one="" angle="" /> 90°), right-angled (one angle = 90°).

Area and Perimeter of a Triangle

Perimeter

The perimeter of a triangle is the total length around it: Perimeter = a + b + c, where a, b and c are the side lengths.

Area

The area of a triangle is half the product of its base and corresponding height:

Area = (Base × Height) ÷ 2

Example: For a triangle with base 4 units and height 6 units:

Area = (4 × 6) ÷ 2 = 12 square units.

Right-Angled Triangles and the Pythagoras Theorem

A right-angled triangle has one angle equal to 90°. The side opposite this right angle is the hypotenuse; the other two sides are called the legs (often called base and perpendicular).

If the legs are a and b, and the hypotenuse is c, then Pythagoras theorem states:

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

∴ a² + b² = c²

This can be understood by drawing squares on each side: area of square on hypotenuse equals sum of areas of squares on the two legs.

Circle

Objects such as coins, bottle caps and wheels suggest the shape of a circle. A circle is a simple closed curve in which every point on the curve is at the same distance from a fixed point called the centre. A circle is not a polygon because it is not made of straight line segments.

Drawing a circle

Common methods to draw a circle:

1. Trace around a circular object such as a coin or bottle cap.
2. Fix one end of a thread at the centre point and tie a pencil to the other end; keeping the thread taut, draw by rotating the pencil.
3. Use a pair of compasses: fix the needle at the centre point and rotate the pencil end keeping the compass open to the required radius.

To draw a circle using compasses:

Step 1: Fix the pencil in the compasses and ensure the needle and pencil tip are at the same level.

Step 2: Place the needle on the paper at the chosen centre.

Step 3: Keep the compass arms fixed at the required opening and rotate the pencil end completely around to draw the circle.

Parts of a circle

• Centre: The fixed point O from which all points on the circle are equidistant. We name a circle by its centre, for example circle O.

• Radius: The line segment joining the centre to any point on the circle; all radii in the same circle are equal. Example: OB is a radius.

• Chord: A line segment joining two points on the circle. Example: XY and MN are chords.

• Diameter: A chord that passes through the centre. A diameter is twice the radius. Example: AB and CD are diameters of circle O.

• Circumference: The length of the boundary of the circle.

Relation between radius and diameter

If AB is a diameter and OA and OB are radii, then AB = 2 × OA = 2 × OB. Thus, diameter = 2 × radius.

Interior, on and exterior of a circle

• Points inside the circle (for example O and P) lie in the interior of the circle.
• Points exactly on the circle (for example M) are on the circle.
• Points outside (for example Q) lie in the exterior of the circle.

To draw a circle of a given radius (for example, radius = 3 cm)

Step 1: Using a ruler, set the compasses to open at 3 cm.

Step 2: Mark a point O on the paper to be the centre.

Step 3: Place the needle of the compasses at O and rotate the pencil around to trace the circle; the resulting circle will have radius 3 cm.

Summary

This chapter presented foundational geometry concepts: points, line segments, lines, rays, planes, types of lines (parallel, intersecting, perpendicular), measurement and drawing of segments and angles, use of a protractor, angle classification, congruent angles, open and closed figures, tangrams as a manipulative tool, properties and classification of triangles, area and perimeter of triangles, Pythagoras theorem for right triangles, and circle definitions and properties (centre, radius, chord, diameter, circumference, interior and exterior). Diagrams and practical activities (ruler, compasses, protractor, tangrams) are emphasised to build geometric intuition and measurement skills useful for classroom teaching.