Triangle || Part 01 || Triangle And Its Parts

Triangle And Its Parts

Triangle :- A polygon with three edges, three vertices and three angles is called Triangles. A triangle with vertices A, B, and C is denoted by ∆ABC.

Note:–

* It is one of the basic shapes in geometry. 

* A Triangle has three angles so it is called Triangle (with three angles).

* A Triangle has three sides so it may be called Trigon. (with three angles).

* Contents of Triangle

A triangle consists of various parts. It has 3 sides, 3 vertices and 3 angles.

1. Vertexes :- The corner or meeting point of the two sides of triangle is called vertexs of triangle.

There are three vertices in a triangle which is denoted by capital letters, A, B and C.

Vertexes

1. Vertex 1 :- A

2. Vertex 2 :- B

3. Vertex 3 :- C

2. Sides :- The line joining the two vertexes of triangle is called Sides of triangle.

There are three Sides in a triangle which is denoted by capital letters, AB or BA, BC or CB and CA or AC.

Sides

1. Side 1 :- AB or BA

2. Side 2 :- BC or CB

3. Side 3 :- CA or AC

Vertex of triangle is denoted by capital letters, A, B and C.

3. Angles :- The depression between two sides at a vertex is called Angle. 

There are three Angles in a triangle which is denoted by the symbol of angle (∠). 

It is denoted in three ways

∠A or AB or BA, AB or BC or CB and AB or CA or AC.

Angles

1. Angle 1 :- ∠A or ∠BAC or ∠CAB

2. Angle 2 :- ∠B or ∠ABC or ∠CBA

3. Angle 3 :- ∠C or ∠ACB or ∠BCA

In this the desired angle is written in between ∠A or ∠BAC or ∠CAB

Triangle:- A polygon with three edges, three vertices and three angles is called Triangles. A triangle with vertices A, B, and C is denoted by ∆ABC.

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Existence of a triangle

A Triangle Is Exist If

* If sum of lengths of any two sides side is greater than third side.

AB + BC > AC

BC + AC > AB

AB + AC > BC

A Triangle is not possible if sum of lengths of any two sides side is less than or equal to third side. 

*1. If sum of lengths of any two sides side is less than to third side. 

AB + BC < AC

BC + AC < AB

AB + AC < BC

*2. If sum of lengths of any two sides side is equal to third side. 

AB + BC = AC

BC + AC = AB

AB + AC = BC

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A Triangle Is Exist If

* If each of the angles is positive.

* If the sum of the all three angles is Supplementary (180°).

∠A + ∠B + ∠C = 180°

* It is also known as Angle Sum Property Of A Triangle so that we can say it in short ASPOT.

* A Triangle is not possible if sum of all the three angles is less than or more  than 180°. 

*1 A Triangle is not possible if sum of all the three angles is less than 180°. 

∠A + ∠B + ∠C < 180°

*2 A Triangle is not possible if sum of all the three angles is more than 180°. 

∠A + ∠B + ∠C > 180°

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Side Angle Relationship

(A). The side opposite to the largest angle of a triangle is the largest and opposite to smallest is smallest.

*1 ∠B is Greater than ∠A & ∠C

then

Side AC opposite to ∠B is Greater than AB & BC

*2 ∠A is Smaller than ∠B & ∠C

then

Side BC opposite to ∠A is smaller than AB & AC

(B) The angle opposite to the largest side of a triangle is the largest and opposite to smallest is smallest.

*1) Side AC opposite to ∠B is Greater than AB & BC

than

∠B is Greater than ∠A & ∠C

*2) Side BC opposite to ∠A is smaller than AB & AC

than

∠A is Smaller than ∠B & ∠C

then

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Exterior Angle Property Of A Triangle (ExAPOT)

Any exterior angle of the triangle is equal to the sum of its interior opposite angles. 

∠ACD = ∠A + ∠B 

     ∠1 = ∠3 + ∠4 

This property of triangle is called the Exterior Angle Property Of A Triangle (ExAPOT).

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SOAEAT

Sum of all exterior angle of the triangle is equal to 360°. 

∠1 + ∠2 + ∠3 = 180°.   [by ASPOT]

∠4 + ∠5 + ∠6 = 360°    [by SOAEAT]

or

∠7 + ∠8 + ∠9 = 360°    [by SOAEAT]

Part 1

∠4 + ∠5 + ∠6 = 360° [by SOAEAT]

Part 2 

∠7 + ∠8 + ∠9 = 360°    [by SOAEAT]

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Special Names Of The Sides Of The Triangle.

Hypotenuse :- Side opposite to right angle is called Hypotenuse. 

We also used this name for opposite side of a given angle. 

Base :- Horizontal side of a triangle is known as base. 

Base is the adjacent side of a given angle.

Perpendicular :- Non-horizontal side of a triangle is known as Perpendicular. 

Perpendicular is the other adjacent side of a given angle.

Perimeter: The sum of the lengths of all the three sides of a triangle is called perimeter.

Perimeter = AB + BC + AC

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Triangle || Part 03 ||  Triangle And Its Types

Triangle:- A polygon with three edges, three vertices and three angles is called Triangles. A triangle with vertices A, B, and C is denoted by ∆ABC.

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(A) Types of Triangles Based on its interior Angles:

Triangles can be classified into three types on the Based of its interior Angles.

Which are:

1. Acute angle or Acute Angle Triangle

2. Obtuse Angle or Obtuse Angle Triangle

3. Right Angle or Right Angle Triangle

1. Acute Triangle or Acute Angled Triangle

A triangle whose all the three interior angles are acute angle (less than 90°) is called Acute Triangle or Acute Angle Triangle. 

Obtuse Triangle or Obtuse Angle Triangle 

A triangle whose one interior angles is obtuse angle (more than 90°) is called Obtuse Triangle or Obtuse Angle Triangle. 

Right Triangle or Right Angle Triangle

A triangle whose one interior angles is right angle (90°) is called Right Triangle or Right Angle Triangle.  

(B) Types of Triangles Based on Sides

On the basis of the sides the triangles can be classified into three types.

Which are:

1. Equilateral Triangle

2. Isosceles Triangle

3. Scalene Triangle

1. Equilateral Triangle or Equiangular Triangle

The triangle whose all the sides are equal in length is called equilateral Triangle. 

It's all three sides are equal in length, so that it's all angles are also equal in measure, which are 60°. 

Since the angles of an equilateral triangle equal in measure, so that it is also known as an equiangular triangle. 

2. Isosceles Triangle

The triangle whose two sides are equal in length is called Isosceles Triangle. 

It's two sides are equal in length, so that it's two angles are also equal in measure.

The opposite angles of equal sides are equal and vice versa.

3. Scalene Triangle

The triangle whose all sides are unequal in length is called scalene Triangle. 

It's all sides are unequal in length, so that it's all three angles are also unequal in measure.

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(C) Types of Triangle Based on Sides and Angles

We classify the triangles further based on their sides and angles as follows:

1. Equilateral or Equiangular Triangle: 

Triangle with all equal sides and equal angles is called an equilateral or equiangular triangle.

2. Acute Isosceles Triangle: 

An Acute Angle Triangle with two equal sides is called an acute isosceles triangle.

3. Isosceles Right Triangle: 

A Right Angle Triangle with two equal sides is called an isosceles right triangle.

4. Obtuse Isosceles Triangle: 

An Obtuse Angle Triangle with two equal sides is called an obtuse isosceles triangle.

5. Acute Scalene Triangle:

An Acute Angle Triangle with unequal sides is called an acute scalene triangle.

6. Right Scalene Triangle: 

A Right Angle Triangle with unequal sides is called an acute Scalene triangle.

7. Obtuse Scalene Triangle: 

An Obtuse Angle Triangle with unequal sides is called an obtuse Scalene triangle.

Ff

Right Angle Triangle And Its Properties

Triangle:- A polygon with three edges, three vertices and three angles is called Triangles. A triangle with vertices A, B, and C is denoted by ∆ABC.

Right Triangle or Right Angle Triangle

A triangle whose one interior angles is right angle (90°) is called Right Triangle or Right Angle Triangle.  

1. Isosceles Right Triangle: 

A Right Angle Triangle with two equal sides is called an isosceles right triangle.

2. Right Scalene Triangle: 

A Right Angle Triangle with unequal sides is called an acute Scalene triangle.

Properties of a Right Angle Triangle

*1) A right angle triangle has a right angle and the other two angles of a right-angle triangle are acute angles.

*2) The side opposite to the right angle is the largest side of the triangle and it is specially named as hypotenuse.

Name Of The Sides Of The Right Angle Triangle

1. Hypotenuse :- Side opposite to right angle is called Hypotenuse. 

We also used this name for opposite side of a given angle. 

2. Base :- Horizontal side of a triangle is known as base. 

Base is the adjacent side of a given angle.

3. Perpendicular :- Non-horizontal side of a triangle is known as Perpendicular. 

Perpendicular is the other adjacent side of a given angle.

Relationship Between The Sides Of The Triangle

Right-Angle Triangle

Pythagoras Theorem

"The square of the hypotenuse (Largest side) is equal to the sum of squares of the other two sides“. This is known as Pythagoras Theorem.

In the ∆ACB :

(AC)² + (BC)² = (AB)²

Special cases of Right Angle Triangles

In 45°-45°-90° Triangle

In this triangle,

First angle is a right angle and the other two angles measure 45° & 45°. This is also called an right isosceles triangle. 

The angles of this triangle are in the ratio – 1: 1: 2, Then sides of this triangle will be in the ratio – 

B : P : H = 1: 1: √2 respectively.

In 30°-60°-90° Triangle

In this triangle,

First angle is a right angle (90°) and the other two angles measure 30° & 60°.

This is also called an right Scalene triangle. 

The angles of this triangle are in the ratio – 1: 2: 3, Then sides of this triangle will be in the ratio – 

B : P : H = 1: √3: 2 respectively.

Pythagorean Triplets

A triple of positive integers (a, b, c) which satisfy the equation a² + b² = c² is called a Pythagorean triple. if a, b, c have no common factor, Such a triple is called primitive triple.

(3, 4, 5), (5, 12, 13), (6, 8, 10), (8, 15, 17), (7, 24, 25), (9, 40, 41), (11, 60, 61)

The formula for Area of Triangle 

Area of any triangle   = ½ × Base × Height