Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Introduction to Euclidean Geometry ↓


Basic Definitions and Terms


Euclidean geometry is a branch of mathematics that investigates the properties and relationships of points, lines, surfaces, and solid figures in space. Developed by the ancient Greek mathematician Euclid, this system of geometry was presented in his famous book, "The Elements." In this comprehensive lesson, we will explore the basic definitions and terms used in Euclidean geometry, providing clear explanations with simple terms and examples to aid understanding.

Point

The point is the most basic unit in geometry. It represents a specific location in space but has no size, width, length, or depth. It is usually represented as a small dot and labeled with a capital letter. For example, a point may be named A.

Example of a point:

A

Line

A line is a straight one-dimensional figure that extends in both directions without end. It has no thickness. Lines are often named using small letters or two points located on them. For example, line AB or line l.

Line l

Example of a line:

A B

Line segment

A line segment is part of a line and has two end points. It is a finite section of a line and includes all the points between its end points. It is named based on its end points. For example, segment AB.

Segment AB

Example of a line segment:

A B

Ray

A ray is a part of a line that starts at a point called the endpoint and extends to infinity in one direction. A ray is named starting from its endpoint followed by another point on the ray. For example, ray AB starts at point A and passes through point B.

Ray AB

Example of a ray:

A B

Plane

A plane is a flat, two-dimensional surface that extends infinitely in all directions. It is often represented in a diagram with a four-sided figure that resembles a tilted square or rectangle. Planes are usually designated by a capital letter or three noncollinear points (points that do not lie on a single line).

Example of plane:

Plane X

Angle

The angle is formed by two rays that share a common end point called the vertex. The rays form the sides of the angle. The angle is often named by three points, with the vertex point listed in the middle, or just by the vertex if it is clear from the context. For example, angle <ABC or <B.

Example of angle:

A B C

Parallel lines

Parallel lines are two or more lines in a plane that do not cross each other, no matter how far they are extended. Parallel lines are always the same distance from each other and are represented by ||. For example, line AB is parallel to line CD, written as AB || CD.

Example of parallel lines:

AB CD

Perpendicular lines

Perpendicular lines are two lines that cross each other at right angles (90 degrees). If two lines are perpendicular, they are represented using ⊥ sign. For example, if line AB is perpendicular to line CD, it can be written as AB ⊥ CD.

Example of perpendicular lines:

AB CD

Circle

A circle is a plane figure where all points are the same distance from a fixed center point. This distance is called the radius. The entire distance around the circle is called the circumference, and the line passing through the center that connects two points on the circle is called the diameter, which is twice the radius.

Circumference = 2π × radius
Diameter = 2 × radius

Example of a circle:

radius Diameter

Polygon

A polygon is a closed figure formed by a limited number of line segments. Polygons are named according to the number of sides they have. Common polygons include triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and hexagons (6 sides).

Example of a polygon:

Triangle Quadrilateral

Summary

Euclidean geometry is foundational to understanding the spatial relationships and properties of various geometric shapes. By mastering these basic terms – points, lines, line segments, rays, angles, planes, parallel and perpendicular lines, circles, and polygons – students build a strong foundation for more complex geometric problem-solving and reasoning. By using examples and simple language, these concepts become easier to understand and apply in both theoretical and practical scenarios.

By learning and visualizing these concepts, you can better understand the world of geometry. Practice drawing different shapes and identifying these components in everyday environments to strengthen your understanding.

Practice problems

  1. Identify and label the points, lines, and angles in the given diagram.
  2. Draw a set of parallel and perpendicular lines and label them.
  3. Sketch polygon examples such as pentagons and hexagons.
  4. Calculate the circumference and diameter of a circle with a given radius of 5 cm.

These exercises will help strengthen your understanding of Euclidean geometry and its basic terms and definitions.


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Basic Definitions and Terms

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