# Properties of Triangle

## Introdcution to Triangle

Triangles are one of the most fundamental and ubiquitous geometric shapes in our world. These three-sided figures are not only visually captivating but also possess a wealth of fascinating properties that have intrigued mathematicians, architects, and scientists for centuries.

At their core, triangles are defined by three distinct sides and three corresponding angles. Regardless of their size or orientation, the sum of the three angles in any triangle always equals 180 degrees. This remarkable characteristic is a testament to the inherent symmetry and balance that lies at the heart of these geometric marvels.

**Also Check: Properties of Rectangle**

### The Basics of Triangles

**Three Sides**: A triangle is a closed shape with three distinct sides, each connected to form a closed loop.**Three Angles**: Correspondingly, a triangle also has three angles, with the sum of these angles always equaling 180 degrees.**Symmetry and Balance**: The unique properties of triangles, such as the constant angle sum, contribute to their remarkable structural integrity and widespread applications in engineering, architecture, and beyond.

## Types of Triangle

**Based on Sides**

- The sides and angles of the Scalene Triangle are all unequal.
- Two equal sides make up an isosceles triangle. In addition, the angles on either side of these equal sides are equal.
- Equilateral Triangle: All three angles are 60 degrees and all three sides are equal.

**Based on angles**

- Acute Angled Triangle: A triangle whose angles are all smaller than 90 degrees.
- A right-angled triangle is one in which one of the three angles is exactly 90 degrees.
- A triangle with one of the three angles greater than 90 degrees is called an obtuse angled triangle.

## Properties of Triangle

**Sum of Angles:** In any type of triangle (whether it is scalene, isosceles, or equilateral), the sum of all three interior angles is always 180 degrees. This is a fundamental property of triangles and holds true for every triangle.

**Side Lengths and Triangle Inequality:** In a triangle, the length of any two sides added together is always greater than the length of the remaining side. This is known as the triangle inequality theorem. For example, if you have a triangle with sides of lengths a, b, and c, then `a + b > c`

, `b + c > a`

, and `c + a > b`

.

**Difference of Sides:** Conversely, the length of any one side of a triangle is always less than the sum of the other two sides and greater than their difference. This means that for sides of lengths a, b, and c, `a < b + c`

, `b < a + c`

, and `c < a + b`

. Similarly, `a > |b - c|`

, `b > |a - c|`

, and `c > |a - b|`

.

**Longest Side and Largest Angle:** In any triangle, the longest side is always opposite the largest angle. This means if you know the largest angle in a triangle, the side opposite this angle will be the longest. For example, in a right-angled triangle, the hypotenuse (the side opposite the right angle) is the longest side.

**Exterior Angle Property:** The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. This is called the exterior angle property. For example, if one of the exterior angles is formed by extending a side of the triangle, this exterior angle is equal to the sum of the two interior opposite angles.

**Similarity of Triangles:** Two triangles are considered similar if their corresponding angles are equal and the lengths of their corresponding sides are in proportion. This means if triangle ABC is similar to triangle DEF, then angle A equals angle D, angle B equals angle E, angle C equals angle F, and the ratios of the lengths of corresponding sides are equal, such as `AB/DE = BC/EF = AC/DF`

.

**Area of a Triangle:** The area of a triangle can be calculated using the formula: `Area = 1/2 * base * height`

. This means you multiply the length of the base of the triangle by the height (the perpendicular distance from the base to the opposite vertex) and then divide by two.

**Perimeter of a Triangle:** The perimeter of a triangle is the total length around the triangle, which is the sum of the lengths of all three sides. If a triangle has sides of lengths a, b, and c, then its perimeter is `a + b + c`

.

#### Related Links

- Derivative of Inverse Trigonometric functions
- Decimal Expansion Of Rational Numbers
- Cos 90 Degrees
- Factors of 48
- De Morgan’s First Law
- Counting Numbers
- Factors of 105
- Cuboid
- Cross Multiplication- Pair Of Linear Equations In Two Variables
- Factors of 100
- Factors and Multiples
- Derivatives Of A Function In Parametric Form
- Factorisation Of Algebraic Expression
- Cross Section
- Denominator
- Factoring Polynomials
- Degree of Polynomial
- Define Central Limit Theorem
- Factor Theorem
- Faces, Edges and Vertices
- Cube and Cuboid
- Dividing Fractions
- Divergence Theorem
- Divergence Theorem
- Difference Between Square and Rectangle
- Cos 0
- Factors of 8
- Factors of 72
- Convex polygon
- Factors of 6
- Factors of 63
- Factors of 54
- Converse of Pythagoras Theorem
- Conversion of Units
- Convert Decimal To Octal
- Value of Root 3
- XXXVII Roman Numerals
- Continuous Variable
- Different Forms Of The Equation Of Line
- Construction of Square
- Divergence Theorem
- Decimal Worksheets
- Cube Root 1 to 20
- Divergence Theorem
- Difference Between Simple Interest and Compound Interest
- Difference Between Relation And Function
- Cube Root Of 1728
- Decimal to Binary
- Cube Root of 216
- Difference Between Rows and Columns
- Decimal Number Comparison
- Data Management
- Factors of a Number
- Factors of 90
- Cos 360
- Factors of 96
- Distance between Two Lines
- Cube Root of 3
- Factors of 81
- Data Handling
- Convert Hexadecimal To Octal
- Factors of 68
- Factors of 49
- Factors of 45
- Continuity and Discontinuity
- Value of Pi
- Value of Pi
- Value of Pi
- Value of Pi
- 1 bigha in square feet
- Value of Pi
- Types of angles
- Total Surface Area of Hemisphere
- Total Surface Area of Cube
- Thevenin's Theorem
- 1 million in lakhs
- Volume of the Hemisphere
- Value of Sin 60
- Value of Sin 30 Degree
- Value of Sin 45 Degree
- Pythagorean Triplet
- Acute Angle
- Area Formula
- Probability Formula
- Even Numbers
- Complementary Angles
- Properties of Rectangle
- Properties of Triangle
- Co-prime numbers
- Prime Numbers from 1 to 100
- Odd Numbers
- How to Find the Percentage?
- HCF Full Form
- The Odd number from 1 to 100
- How to find HCF
- LCM and HCF
- Calculate the percentage of marks
- Factors of 15
- How Many Zeros in a Crore
- How Many Zeros are in 1 Million?
- 1 Billion is Equal to How Many Crores?
- Value of PI
- Composite Numbers
- 100 million in Crores
- Sin(2x) Formula
- The Value of cos 90°
- 1 million is equal to how many lakhs?
- Cos 60 Degrees
- 1 Million Means
- Rational Number
- a3-b3 Formula with Examples
- 1 Billion in Crores
- Rational Number
- 1 Cent to Square Feet
- Determinant of 4×4 Matrix
- Factor of 12
- Factors of 144
- Cumulative Frequency Distribution
- Factors of 150
- Determinant of a Matrix
- Factors of 17
- Bisector
- Difference Between Variance and Standard Deviation
- Factors of 20
- Cube Root of 4
- Factors of 215
- Cube Root of 64
- Cube Root of 64
- Cube Root of 64
- Factors of 23
- Cube root of 9261
- Cube root of 9261
- Determinants and Matrices
- Factors of 25
- Cube Root Table
- Factors of 28
- Factors of 4
- Factors of 32
- Differential Calculus and Approximation
- Difference between Area and Perimeter
- Difference between Area and Volume
- Cubes from 1 to 50
- Cubes from 1 to 50
- Curved Line
- Differential Equations
- Difference between Circle and Sphere
- Cylinder
- Difference between Cube and Cuboid
- Difference Between Constants And Variables
- Direct Proportion
- Data Handling Worksheets
- Factors of 415
- Direction Cosines and Direction Ratios Of A Line
- Discontinuity
- Difference Between Fraction and Rational Number
- Difference Between Line And Line Segment
- Discrete Mathematics
- Disjoint Set
- Difference Between Log and Ln
- Difference Between Mean, Median and Mode
- Difference Between Natural and whole Numbers
- Difference Between Qualitative and Quantitative Research
- Difference Between Parametric And Non-Parametric Tests
- Difference Between Permutation and Combination

## Frequently Asked Questions on Properties of Triangle

The 7 key properties of a triangle are: 1) Three sides, 2) Three angles, 3) The sum of the angles is 180 degrees, 4) The longest side is opposite the largest angle, 5) The shortest side is opposite the smallest angle, 6) Two sides are always longer than the third side, and 7) A triangle can be classified into different types based on its sides and angles.

The 3 main properties of a right triangle are: 1) One angle is a 90-degree right angle, 2) The two sides meeting at the right angle are called the legs, and 3) The side opposite the right angle is called the hypotenuse, which is the longest side.

The three triangle property states that the sum of the angles in any triangle is always 180 degrees. This means that the three angles in a triangle, when added together, will always equal 180 degrees.

The 45-45-90 triangle theorem states that in a right triangle where two angles are 45 degrees each, the third angle is 90 degrees (a right angle). Additionally, the two legs of the triangle are equal in length.

A 33-degree angle is called an acute angle. An acute angle is any angle that is less than 90 degrees. In a triangle, the three angles can be a combination of acute, obtuse, and right angles, depending on their specific measurements.