Geometry is an important branch of mathematics that involves studying various shapes, figures, dimensions, angles, etc. It provides formulas to calculate the area, perimeter, circumference, and volume of various figures. Geometrical figures are divided into two types based on dimensions:

1. 2D figures: Shapes with only 2 dimensions, length and breadth, are called 2D figures. We can calculate the area and perimeter of 2D figures. Example: triangles, circles, etc.
2. 3D figures: Shapes with 3 dimensions, length, breadth, and height, are called 3D figures. We can calculate the surface area and volume of 3D figures. Example: cones, cubes, cylinders, etc.

Triangles are three-sided 2D figures that satisfy the following conditions:

• The sum of any two sides is greater than the third side of the figure.
• The angle exactly opposite to the longest side is the largest angle of the figure.
• The total sum of the interior angles of the figure is exactly 180°

Area of triangle = ½ × (b × h), where b is base and h is the height of the triangle

The perimeter of a triangle = Sum of all sides

Types of Triangles

Triangles are differentiated based on their angles and sides.

Based on the angles:

• Acute triangle: Each angle of the triangle is less than 90°.
• Right-angled triangle: One angle of the triangle is equal to 90°.
• Obtuse triangle: One angle of the triangle is greater than 90° but less than 180°.
• Equiangular triangle: Each angle of the triangle is equal to 60°.

Based on the sides:

• Scalene triangle: The lengths of all sides are different.
• Isosceles triangle: The lengths of any two sides are equal.
• Equilateral triangle: All sides of the triangle are equal.

Congruent Triangles 

Two triangles are said to be congruent if both are of the same shape and size. For example, two equilateral triangles of side 6 cm are congruent to each other.

Similar Triangles 

Two triangles are said to be similar if both are of the same shape but different sizes. For example, two right-angled triangles are similar to each other but might not be congruent.

Also known as Thales’s theorem, it states that if a line DE is drawn parallel to the base BC of triangle ABC, then AD/DB = AE/EC.

This theorem applies only to right-angled triangles where one angle is equal to 90°. The longest side of the right-angled triangle is opposite the 90° angle and is known as hypotenuse.

According to this theorem, the square of the hypotenuse of a right-angled triangle is always equal to the sum of the squares of the other two sides of the triangle. 

Example: In a right-angled triangle ABC, where AB is the hypotenuse, AB^2 = BC^2 + AC^2.

Closed geometrical figures made of straight lines are known as rectilinear figures. In layman’s terms, closed figures made of at least three line segments are known as polygons or rectilinear figures. 

Polygons are classified based on the sides:

• 3-sided figures: Triangles 
• 4-sided figures: Quadrilateral 
• 5-sided figures: Pentagon 
• 6-sided figures: Hexagon 
• 7-sided figures: Heptagon 
• 8-sided figures: Octagon 

A regular polygon is a figure where all sides, exterior angles, and interior angles are equal.

The following are the formulas for polygons with side ‘n’:

• The sum of interior angles is equal to (2n-4) × 90°
• The sum of all the exterior angles of the polygon is equal to 360°
• The exterior angle of a regular polygon is equal to 360°/n, whereas an interior angle of a regular polygon is equal to (2n-4) × 90°/n.
• The sum of an exterior angle and interior angle is 180° at every vertex of the triangle.

Circles are 2D figures measured on the basis of the radius. As it is a 2D shape, it has both area and perimeter. 

Area = πr^2

Circumference (Perimeter) = 2πr

Properties of Circle 

• The angle of a semicircle is 90°.
• The angles from a chord in the same segment are equal.
• The central angle of the circle is twice the angle on the circumference of the circle.
• The perpendicular line drawn from the centre of the circle divides the chord into two equal parts.

This chapter is an important topic for class 9th mathematics and an essential concept for advanced and higher-level mathematics. Sound knowledge of this chapter will prepare you for advanced lessons.

1. What do you mean by geometry?
   Geometry is a branch of mathematics that involves studying various shapes, dimensions, angles, etc.
2. What is the formula to calculate the sum of interior angles of a polygon?
   The sum of interior angles is equal to (2n-4) × 90°
3. What is a cyclic quadrilateral?
   A cyclic quadrilateral is a quadrilateral inside a circle.
4. What is the property of a cyclic quadrilateral?
   The sum of the opposite angles of a cyclic quadrilateral is 180°.
5. Are all congruent triangles similar?
   Yes, all congruent triangles are similar, but the opposite is not true.