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 → Lines and Angles ↓


Types of Angles


Angles are one of the fundamental concepts in geometry. Understanding angles and their types helps us understand shapes, measure shapes, and even construct shapes. In geometry, an angle is formed when two rays or lines meet at a common point called the vertex. The amount of rotation between each side of the angle is measured in degrees.

Different types of angles

Angles can be classified in several ways depending on their measure and their relationship to other angles. Below, we will discuss the different types of angles in detail with explanations, examples, and diagrams.

1. Acute angle

An acute angle measures less than 90 degrees. It looks sharp and narrow. You can find acute angles in a variety of everyday objects, such as the blades of scissors when they are partially open.

Example: If the measure of an angle is 45 degrees, it is considered an acute angle.

acute angle

2. Right angle

A right angle measures exactly 90 degrees. It represents a quarter turn and is often seen in the corners of squares and rectangles.

Example: The angle between the x-axis and y-axis on a graph is a right angle.

Right angle

3. Obtuse angle

An obtuse angle is an angle whose measure is more than 90 degrees but less than 180 degrees. It appears wider than a right angle.

Example: If the measure of an angle in a triangle is 130 degrees, then it is an obtuse angle. Such triangles are called obtuse-angled triangles.

obtuse angle

4. Straight angle

A straight angle is exactly 180 degrees. It looks like a straight line, indicating a half turn.

Example: The angle between the two hands of a clock at 6 o'clock is a straight angle.

straight angle

5. Reflex angle

A reflex angle is greater than 180 degrees but less than 360 degrees. Reflex angles appear in situations where there is more than a straight angle but less than a complete rotation.

Example: When the time is 10 o’clock then the larger angle made by the hands of a clock is the reflex angle.

reflex angle

6. Full angle

A complete angle is 360 degrees, which means it makes a complete circle. When an object moves in a complete circle, it makes a complete angle.

Example: One complete revolution of a wheel makes a complete angle.

Full angle

Complementary and supplementary angles

In addition to individual angle measurements, angles can also be classified based on their relationship with other angles.

Supplementary angles

Two angles are said to be complementary if the sum of their measures is 90 degrees. These angles are often seen in right triangles, where the other two angles (except the right angle) are supplementary.

Example: If the measure of one angle is 30 degrees, then the measure of the other angle must be 60 degrees to be complementary.

(Angle 1) + (Angle 2) = 90°

Obtuse angle

Two angles are considered supplementary if their combined measures equal 180 degrees. These are usually seen along a straight line.

Example: If one angle is 110 degrees, then the measure of the other angle will be 70 degrees, so that the two angles are supplementary.

(Angle 1) + (Angle 2) = 180°

Visualization of angle relationships

When working with angles, it is important to see how they relate visually within geometric shapes or configurations. Below are some examples of how to represent these relationships:

45° 45° Supplementary angles

In the figure above, both of the smaller angles are 45 degrees, making them complementary since their sum is 90 degrees.

Angles in triangles

Triangles are geometric shapes that have three sides and three angles. The sum of the angles in a triangle is always 180 degrees. Here are some types of triangles:

Equilateral triangle

  • All three interior angles are equal (60 degrees each).
60° 60° 60°

Isosceles triangle

  • The two interior angles are equal.
50° 65° 65°

Scalene triangle

  • All interior angles are different.
40° 60° 80°

By understanding and visualizing the different types of angles and their relationships, we gain a deeper understanding of the geometric world around us. This knowledge is the foundational skill that enables us to explore more complex geometric concepts.


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Types of Angles

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