Classification of Triangles, Circles and Quadrilaterals

Introduction

Geometry, one of the fundamental branches of mathematics, deals with the study of shapes, sizes and properties of objects in space. Among the various geometric figures, triangles, circles and quadrilaterals hold a special place due to their prevalence and importance in both mathematics and the real world. In this article, we will delve into the classification of these three shapes, exploring their properties and characteristics.

Triangles

A triangle is a polygon formed by three straight-line segments. Based on their sides and angles, triangles can be classified into several types:

Scalene Triangle

In a scalene triangle, all three sides and three angles are different from each other, making them unequal.

Isosceles Triangle

An isosceles triangle is characterized by having two sides of equal length and two angles of equal measure.

Equilateral Triangle

An equilateral triangle has three equal sides and three equal angles, each measuring 60^o.

Angle of Triangles

Interior Angles

The interior angles of a triangle are the angles that are formed within the triangle. Let's denote the three interior angles as ∠A, ∠B and ∠C, corresponding to the vertices A, B and C, respectively. The sum of these interior angles is always equal to 180^o, known as the angle sum property of a triangle.

Therefore, we have the equation: ∠A + ∠B + ∠C = 180°

Exterior Angles

An exterior angle of a triangle is formed by extending one of its sides outward. Each vertex of a triangle has a corresponding exterior angle. Let's denote the exterior angles as ∠D, ∠E and ∠F, corresponding to the vertices A, B and C, respectively. The sum of the exterior angles of any triangle is always 360^o.

This can be expressed as: ∠D + ∠E + ∠F = 360°

An important observation to make is that the measure of an exterior angle at any vertex of a triangle is equivalent to the sum of the two interior angles that are not adjacent to it.
For example, ∠D = ∠B + ∠C, ∠E = ∠A + ∠C and ∠F = ∠A + ∠B.

Relationships between Interior and Exterior Angles

The interior and exterior angles of a triangle have an interesting relationship. The sum of a specific interior angle and its corresponding exterior angle is always 180^o. In other words:
∠A + ∠D = ∠B + ∠E = ∠C + ∠F = 180°

Triangles can be classified into three categories based on the measurement of their angles.

Right Triangle: A right triangle contains one right angle, which measures 90^o.

Acute Triangle: An acute triangle has three acute angles, all measuring less than 90^o.

Obtuse Triangle: An obtuse triangle has one obtuse angle, which measures more than 90^o.

Circles

A circle is a two-dimensional geometric figure that is perfectly round. It is defined by a set of points equidistant from a central point called the centre.

Here are some key terms associated with circles:

Centre

The point in the middle of the circle from which all points on the circle are equidistant.

Radius

The distance from the centre of the circle to any point on its circumference.

Diameter

The line segment that passes through the centre of the circle and has its endpoints on the circle. The diameter is twice the length of the radius.

Circumference

The distance around the outer boundary of the circle. It is calculated using the formula: circumference = 2πr, where r is the radius.

Chord

A line segment that connects two points on the circumference of the circle.

Arc

A curve is a part of the circumference of a circle.

Sector

A region bounded by two radii of a circle and the arc connecting them.

Segment

A segment is a region that is enclosed by a chord and the arc between the endpoints of the chord. This region does not include the centre of the circle.

Quadrilaterals

A quadrilateral is a polygon with four sides. It is classified based on the length of its sides and the measure of its angles.

The following are some common types of quadrilaterals:

Square

A square is a quadrilateral with four equal sides and four right angles and bisects each other at 90^o.

Rectangle

A rectangle is a quadrilateral with four right angles, but the opposite sides are equal in length and bisect each other at 90^o.

Rhombus

A rhombus is a quadrilateral with all sides equal in length, but the opposite angles are not necessarily right angles.

Parallelogram

A parallelogram is a quadrilateral in which the opposite sides are parallel to each other.

Trapezium

A trapezium is a four-sided polygon characterized by having at least one pair of parallel sides.

Kite

A kite is a quadrilateral characterized by having two pairs of adjacent sides that are of equal length.

Sides and Angles of Quadrilateral