Triangles in geometry( Explanation, types, examples and questions).

Triangle:-

A triangle is a polygon with three sides and three angles. It is one of the basic shapes in geometry. The sum of the angles in a triangle is always 180 degrees. Triangles can have different classifications based on the lengths of their sides and the measures of their angles.

Table of contents:- 

1. Who discovered triangle? 
2. Types of triangles.
• On the basis of angle.
• On the basis of sides.
3. Properties of triangles.
4. Theorem.
5. Perimeter.
6. Areas.

Who discovered triangle ?

The triangle was known in China in the early 11th century by the mathematician Jia Xian (1010–1070).

Types of triangles :-

There are two types of triangles.

a. On the basis of angles :-

i. Acute angle triangles:-

 A triangle with all three angles measuring less than 90 degrees.

Here, ∆ABC is an acute angle triangle.

ii. Right angle triangles:-

A right triangle, also known as a right-angled triangle, is a type of triangle that has one angle measuring 90 degrees (a right angle). The other two angles are acute angles, meaning they are less than 90 degrees.

 

iii. Obtuse Triangle: 

An obtuse triangle has one angle measuring more than 90 degrees (an obtuse angle). The other two angles are acute.

Here, ∆ABC is an obtuse angle triangle.

b. On the basis of sides:-

i. Equilateral triangles:

An equilateral triangle is a type of triangle in which all three sides are equal in length, and all three angles are equal to 60 degrees.

 

 This ∆ is equilateral triangle.

ii. Isosceles triangle:-

An isosceles triangle is a type of triangle that has at least two sides of equal length. In an isosceles triangle, the angles opposite the equal sides are also equal. The third side, called the base, is typically of a different length than the equal sides.

Here, it is an isosceles triangle.

iii.Scalene Triangle:

 A scalene triangle has all three sides of different lengths. None of the angles in a scalene triangle are equal. Each angle has a different measure.

Here, ∆ABC is scalene triangle.

Properties of triangle:-

Note:- for all properties of triangles, click here

i. Angle sum property:-

The sum of the three interior angles of a triangle is always 180 degrees. This property is known as the Angle Sum Property of a Triangle.

Here, In ∆ABC 

         angle A + angle B + angle C = 180°.

ii. Exterior angle property:-

The exterior angle property of a triangle states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles. In other words, if you extend one side of a triangle, the angle formed by this extension is called an exterior angle.

 Here, angle e = angle a + angle b.

iii. Sum of all sides of a triangle is always 180°.

iv. Sum of all exterior angle of triangles is always 360°.

v. All equilateral triangles are similar.

vi. In triangle equal oposite side have equal oposite angle.

Vii. Equilateral triangles have all angle of 60°

Viii. Triangle is a closed polygon having 3 side.

Theorem related to triangle:- 

i. Pythagoras theorem:- It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, the Pythagorean theorem can be expressed as:

a² + b² = c²

ii. Equal oposite side have equal oposite angle:-

The theorem you are referring to is known as the Converse of the Isosceles Triangle Theorem or the Converse of Base Angles Theorem. It states that if two sides of a triangle are congruent (equal in length), then the angles opposite those sides are congruent (equal in measure).

In symbolic notation, if we have a triangle with sides of lengths "a," "b," and "c," and angles A, B, and C opposite those sides, the theorem can be stated as follows:

If a = b, then angle A = angle B.

Similarly, if b = c, then angle B = angle C, and if a = c, then angle A = angle C.

This theorem is based on the concept of the properties of isosceles triangles. An isosceles triangle has at least two sides of equal length, which leads to the conclusion that the angles opposite those sides are also equal.

The Converse of the Isosceles Triangle Theorem is a useful tool in triangle geometry and is often employed to prove various results and solve problems involving congruent angles in triangles. 

 Perimeter of a triangle:- 

The perimeter of a triangle is the total length of all its sides. To calculate the perimeter, you simply need to add the lengths of the three sides together.

Let us assume a ∆ABC in which AB = 10 cm, BC = 5 cm and CA= 2 cm 

Then, the perimeter be AB +BC+CA 

i.e., 10+5+2 = 17cm

Area of triangles:- 

The area of a triangle can be calculated using different formulas depending on the given information about the triangle. Here are three common methods for calculating the area of a triangle:

1. Using Base and Height:

If you know the length of the base (b) of the triangle and the perpendicular height (h) from that base to the opposite vertex, you can calculate the area (A) using the formula:

A = (1/2) * b * h

2. Using Heron's Formula:

If you know the lengths of all three sides of the triangle, you can use Heron's Formula to calculate the area. Let the lengths of the sides be a, b, and c, and let s be the semiperimeter of the triangle, which is calculated as half the sum of the lengths of the sides:

s = (a + b + c) / 2

Then, the area (A) of the triangle can be calculated using the formula:

A = √(s * (s - a) * (s - b) * (s - c))

Heron's Formula is particularly useful when you don't have the height of the triangle but have the lengths of all three sides.

3. Using Base and Side Length:

If you know the length of the base (b) and one of the side lengths (a) of the triangle, along with the included angle (θ) between those two sides, you can use the following formula to calculate the area (A):

A = (1/2) * a * b * sin(θ)

This formula uses the trigonometric function sine to calculate the area.

It's important to use the appropriate formula based on the available information about the triangle.