Class 9 Mathematic Lines and Angles All Formula & Concept Notes | Formula Sheet

Class 9 Mathematics covers various fundamental concepts and formulas related to Lines and Angles. Here’s a detailed view:

1. Basic Definitions

• Line: A straight one-dimensional figure that extends infinitely in both directions.
• Line Segment: A part of a line with two endpoints.
• Ray: A part of a line that starts at one point and extends infinitely in one direction.
• Angle: Formed when two rays originate from the same endpoint. The common endpoint is called the vertex, and the rays are the arms of the angle.

2. Types of Angles

• Acute Angle: An angle less than 90°.
• Right Angle: An angle equal to 90°.
• Obtuse Angle: An angle greater than 90° but less than 180°.
• Straight Angle: An angle equal to 180°.
• Reflex Angle: An angle greater than 180° but less than 360°.
• Complete Angle: An angle equal to 360°.

3. Complementary and Supplementary Angles

• Complementary Angles: Two angles whose sum is 90°.
• Supplementary Angles: Two angles whose sum is 180°.

4. Adjacent Angles

• Angles that have a common vertex and a common arm but do not overlap.

5. Linear Pair

• A pair of adjacent angles whose non-common arms are opposite rays. The sum of the angles in a linear pair is 180°.

6. Vertically Opposite Angles

• When two lines intersect, the angles opposite to each other are called vertically opposite angles. They are equal.

7. Parallel Lines and Transversal

• Parallel Lines: Lines that never meet, no matter how far they are extended.
• Transversal: A line that intersects two or more lines at distinct points.

8. Angles Formed by a Transversal

• Corresponding Angles: Angles that are in the same position on the two lines in relation to the transversal. They are equal if the lines are parallel.
• Alternate Interior Angles: Angles that lie between the two lines on opposite sides of the transversal. They are equal if the lines are parallel.
• Alternate Exterior Angles: Angles that lie outside the two lines on opposite sides of the transversal. They are equal if the lines are parallel.
• Consecutive Interior Angles (Co-interior Angles): Angles that lie on the same side of the transversal and inside the two lines. They are supplementary if the lines are parallel.

9. Angle Sum Property of a Triangle

• The sum of the interior angles of a triangle is 180°.
• Exterior Angle Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

10. Properties of Special Triangles

• Equilateral Triangle: All angles are 60°.
• Isosceles Triangle: The angles opposite the equal sides are equal.
• Scalene Triangle: All angles are different.

11. Properties of Special Quadrilaterals

• Parallelogram: Opposite angles are equal, and consecutive angles are supplementary.
• Rectangle: All angles are 90°, and opposite sides are equal.
• Rhombus: All sides are equal, and opposite angles are equal.
• Square: All sides are equal, and all angles are 90°.
• Trapezium: Only one pair of opposite sides is parallel.

12. Angle Bisector

• A line that divides an angle into two equal parts.

13. Perpendicular Bisector

• A line that is perpendicular to a segment and divides it into two equal parts.

14. Important Theorems

• Theorem 1: If two lines intersect, the vertically opposite angles are equal.
• Theorem 2: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
• Theorem 3: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the lines are parallel.
• Theorem 4: The sum of the angles in a linear pair is 180°.
• Theorem 5: The sum of the angles of a triangle is 180°.

15. Practical Applications

• Parallelism in Real Life: Roads, railway tracks, etc., represent parallel lines.
• Angles in Design: Architecture, art, and engineering designs often utilize different types of angles.

16. Formulas to Remember

• Angle Sum Property of a Triangle: $\angle A + \angle B + \angle C = 180^\circ$
• Exterior Angle of a Triangle: $\text{Exterior Angle} = \angle A + \angle B$ (where A and B are the non-adjacent interior angles)
• Angle Sum Property of Quadrilateral: $\angle A + \angle B + \angle C + \angle D = 360^\circ$
• Linear Pair: $\angle A + \angle B = 180^\circ$
• Supplementary Angles: $\angle A + \angle B = 180^\circ$
• Complementary Angles: $\angle A + \angle B = 90^\circ$

17. Practice Problems

• Solve problems involving the calculation of unknown angles using the above properties.
• Use the properties of parallel lines and transversals to prove angles are equal or supplementary.
• Apply the Angle Sum Property to find unknown angles in triangles and quadrilaterals.

By mastering these concepts, students will be able to solve a variety of problems related to lines and angles, both theoretically and practically