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


Euclidean geometry is a branch of mathematics that deals with shapes, lines, angles, and surfaces. It is named after the ancient Greek mathematician Euclid, who wrote a book called "Elements" over 2,000 years ago. In this book, Euclid laid out the fundamentals and principles of geometry that we still use today. This introduction is intended to provide a comprehensive and easy-to-understand guide to the basic concepts in Euclidean geometry for grade 9 students.

Basic concepts

Let's start with some basic concepts of geometry. These are the cornerstones that will help you understand more complex ideas later.

Point

A point is a location in space. It has no size, no width, no depth - it just represents a position. We represent a point by a dot and usually name it with a letter, like A or B

A

Line

A line is a straight one-dimensional figure that has no thickness and extends infinitely in both directions. Lines are usually named by small letters or two points on the line, such as line AB.

A B

Line segment

A line segment is a portion of a line that has two end points. Unlike a line, a line segment does not extend to infinity. We represent a line segment by its end points, for example, line segment AB.

A B

Ray

A ray starts at a point and goes to infinity in a particular direction. A ray is usually named by the name of its end point and another point on the ray. For example, ray AB starts at A and passes through B

A B

Plane

A plane is a flat two-dimensional surface that extends infinitely in all directions. Imagine it as a giant, flat sheet of paper with no end.

Angles

Angles are formed when two lines or line segments meet at a common point, called the vertex. Angles are measured in degrees.

Types of angles

  • Acute Angle: An angle less than 90 degrees.
  • Right Angle: An angle of exactly 90 degrees.
  • Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.
  • Straight angle: An angle of exactly 180 degrees.

When drawing angles, we usually represent them with a circular arc and a label, such as ∠ABC.

A B C

Triangle

Triangles are polygons made up of three line segments. The types of triangles can be determined by their angles or the lengths of their sides.

Classification based on angles

  • Acute Triangle: All angles are less than 90 degrees.
  • Right Angle: An angle exactly 90 degrees.
  • Obtuse Triangle: One of its angles is more than 90 degrees.

Classification based on sides

  • Equilateral Triangle: All sides are of equal length.
  • Isosceles triangle: Two sides are of equal length.
  • Scalene Triangle: All sides are of different lengths.
Equilateral Isosceles scalene

Quadrilateral

Quadrilaterals are polygons with four sides. Some common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, and trapezoids.

Types of quadrilaterals

  • Square: All sides are equal, and all angles are 90 degrees.
  • Rectangle: Opposite sides are equal, and all angles are 90 degrees.
  • Parallelogram: Opposite sides are parallel and equal in length.
  • Rhombus: All sides are equal, and opposite angles are also equal.
  • Trapezoid: It has only one pair of opposite sides parallel.
Social class rectangle

Congruence and similarity

Congruence and similarity are important concepts in geometry that deal with the comparison of geometric shapes.

Conformity

Two figures are congruent if they have the same shape and size. Congruent figures can be transformed into each other through rigid motions such as translation, rotation or reflection. When two triangles are congruent, we write △ABC ≅ △DEF.

Equality

Two figures are similar if they have the same shape but not necessarily the same size. Similar figures can be transformed into each other through stretching in addition to rigid motion. When two triangles are similar, we write △ABC ~ △DEF.

Pythagorean Theorem

The Pythagorean theorem is a fundamental theorem in Euclidean geometry. It deals with the lengths of the sides of a right-angled triangle. The theorem states:

a² + b² = c²

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Summary

Understanding Euclidean geometry is essential because it forms the basis for studying more complex mathematical concepts. This introduction covers the basic elements: points, lines, angles, shapes, congruence, and similarity. By mastering these topics, you gain the skills to solve a variety of geometric problems.


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Introduction to Euclidean Geometry

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