Properties and Types of Triangles

Triangles

Properties of Triangles

• Sum of interior angles: 180°
• Exterior angle is equal to the sum of two opposite interior angles
• Sum of any two sides is greater than the third side
• If two angles are equal, then the sides opposite to them are also equal

Types of Triangles

• Equilateral Triangle: all sides are equal
• Isosceles Triangle: two sides are equal
• Scalene Triangle: all sides are unequal
• Right Angled Triangle: one angle is 90°
• Oblique Triangle: no angle is 90°

Congruent and Similar Triangles

• Congruent Triangles: identical in shape and size
• Similar Triangles: identical in shape but not in size
• Criteria for Congruence: SAS, ASA, SSS, RHS
• Criteria for Similarity: AA, SSS, SAS

Linear Equations

Introduction

• Linear Equation in one variable: ax + b = 0, where a ≠ 0
• Linear Equation in two variables: ax + by + c = 0, where a, b ≠ 0

Solution of Linear Equations

• Method of Substitution
• Method of Elimination
• Graphical Method

Pair of Linear Equations in Two Variables

• Consistent System: has a unique solution
• Inconsistent System: has no solution
• Dependent System: has infinitely many solutions
• Independent System: has a unique solution

Graph of Linear Equation

• Slope-Intercept Form: y = mx + c, where m is slope and c is y-intercept
• Intercept Form: x/a + y/b = 1, where a, b are x, y intercepts respectively

Surface Areas

Surface Area of 3D Shapes

• Cube: 6l^2, where l is side length
• Cuboid: 2(lb + bh + lh), where l, b, h are length, breadth, height respectively
• Cylinder: 2πr(r + h), where r is radius and h is height
• Cone: πr(r + l), where r is radius and l is slant height
• Sphere: 4πr^2, where r is radius

Formulae

• Lateral Surface Area: LSA = πr(l), where r is radius and l is slant height
• Total Surface Area: TSA = 2πr(r + h), where r is radius and h is height

Triangles

Properties of Triangles

• The sum of interior angles in a triangle is always 180°
• An exterior angle is equal to the sum of two opposite interior angles
• The sum of any two sides of a triangle is greater than the third side
• If two angles in a triangle are equal, then the sides opposite to them are also equal

Types of Triangles

• An equilateral triangle has all sides of equal length
• An isosceles triangle has two sides of equal length
• A scalene triangle has all sides of unequal length
• A right-angled triangle has one angle of 90°
• An oblique triangle has no angle of 90°

Congruent and Similar Triangles

• Congruent triangles are identical in shape and size
• Similar triangles are identical in shape but not in size
• The criteria for congruence are SAS, ASA, SSS, and RHS
• The criteria for similarity are AA, SSS, and SAS

Linear Equations

Introduction

• A linear equation in one variable is of the form ax + b = 0, where a ≠ 0
• A linear equation in two variables is of the form ax + by + c = 0, where a, b ≠ 0

Solution of Linear Equations

• The method of substitution is used to solve linear equations
• The method of elimination is used to solve linear equations
• The graphical method is used to solve linear equations

Pair of Linear Equations in Two Variables

• A consistent system has a unique solution
• An inconsistent system has no solution
• A dependent system has infinitely many solutions
• An independent system has a unique solution

Graph of Linear Equation

• The slope-intercept form of a linear equation is y = mx + c, where m is slope and c is y-intercept
• The intercept form of a linear equation is x/a + y/b = 1, where a, b are x, y intercepts respectively

Surface Areas

Surface Area of 3D Shapes

• The surface area of a cube is 6l^2, where l is side length
• The surface area of a cuboid is 2(lb + bh + lh), where l, b, h are length, breadth, height respectively
• The surface area of a cylinder is 2πr(r + h), where r is radius and h is height
• The surface area of a cone is πr(r + l), where r is radius and l is slant height
• The surface area of a sphere is 4πr^2, where r is radius

Formulae

• The lateral surface area (LSA) of a cone is πr(l), where r is radius and l is slant height
• The total surface area (TSA) of a cylinder is 2πr(r + h), where r is radius and h is height