Properties of Triangle

A triangle is a form having three sides and three corners. Each side links to two adjacent sides, resulting in three corners where the sides meet. The angles within a triangle always sum to 180 degrees. Triangles are classified into three types, namely equilateral (all sides and angles are equal), isosceles (two sides and two angles are equal), and scalene (all sides and angles differ).

In this article, we will understand the concepts related to triangle: meaning and definition of tringle, properties of triangle, types of triangle and formulas of triangle.

What is a Triangle?

Triangle is a basic geometric form that has three sides and three angles. Imagine connecting three points on a level area using straight lines to make a triangle. Each side of the triangle links two vertices, and an angle is generated when two sides intersect.

Consider a piece of pizza that has three sides and three corners, similar to a triangle. Another example is a street sign with three edges that intersect at each corner. Triangles are fundamental forms seen in many common items, and they are required in geometry to comprehend the basic concepts of angles and measures.

Triangle Definition

A triangle is a fundamental geometric form consisting of three straight sides and three angles. The sides meet to form three corners. The total of the angles within a triangle is always 180 degrees. Triangles have varied qualities; for example, they can be equilateral (all sides and angles equal), isosceles (two sides and angles equal), or scalene.

What are Properties of Triangle?

Before getting into the properties of triangles, it’s important to understand their basic structure and characteristics:

• A triangle is composed of three sides, three angles, and three vertices.
• Sum of all internal angles is always 180°.
• Sum of the lengths of any two sides exceeds the length of the third.
• Longest side corresponds to the biggest angle.
• Exterior angle is equal to sum of opposite two interior angle.

Types of Triangles

Triangles are classified according to their side lengths and angles. Here are the six primary categories explained simply:

Equilateral Triangle: An equilateral triangle has all sides of equal length and angles of 60 degrees. For example, consider a triangle with all three sides measuring five units long.

Isosceles Triangle: An isosceles triangle has two sides of equal length and opposing angles. For example, consider a triangle with two sides measuring four units apiece and a third side measuring six units.

Scalene Triangle: A scalene triangle has sides of varying lengths and angles of varying degrees. Consider a triangle with sides of 3, 4, and 5 units.

Acute Triangle: Acute triangles have all angles smaller than 90 degrees. For example, a triangle with angles of 30, 60, and 80 degrees.

Right Triangle: A right triangle has one angle that measures exactly 90 degrees, which is known as the right angle. Consider a triangle with angles of 30, 60, and 90 degrees.

Obtuse Triangle: An obtuse triangle has one angle greater than 90 degrees. For example, a triangle with angles measuring 30, 60, and 120 degrees.

Types of Triangles

Properties of Triangle

Properties of triangles are fundamental rules in geometry. They include the angle sum property, triangle inequality property, Pythagoras theorem, side-angle relationship, exterior angles, and congruence conditions, etc.

Some of the important properties of triangle are added below:

Angle Sum Property

Angle Sum Property is a fundamental property in geometry that asserts that the sum of all angles within a triangle is always 180 degrees. This technique is useful for solving for missing angles or determining triangle validity. For example, if two angles are 60 degrees each, the third angle must also be 60 degrees to meet this criterion.

Angle 1 + Angle 2 + Angle 3 = 180^∘

Triangle Inequality Property

The total of any two sides of a triangle exceeds the length of the third side. In other words, the shortest path between two places is a straight line. This is expressed as:

a + b > c

where a, b, and c are the lengths of the sides of the triangle.

Pythagoras Property

In a right triangle, the square of the hypotenuse’s length (the side opposite the right angle) equals the sum of the squares of the other two sides. This is called the Pythagorean theorem.

c² = a² + b²

Hypotenuse length is denoted by c, whereas the other two sides’ lengths are denoted by a and b.

Side Opposite the Greater Angle is the Longest Side

The side opposite the greatest angle in a triangle is the longest side. This is an observable property, not a formal theorem. When given a triangle’s angles, it helps to determine which side is the longest.

Exterior Angle Property

Each exterior angle of a triangle equals the sum of its two remote interior angles. The mathematical expression is:

Exterior Angle = Sum of Remote Interior Angles

Congruence Property

Triangles of the same size and shape are said to be congruent. This attribute is useful in assessing if two triangles are identical. It may be proved by congruence criteria such as

• Side-Side-Side (SSS)
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Right Angle-Hypotenuse-Side(RHS)

These qualities are essential for understanding and solving triangle-related geometry issues.

Formulas of Triangle

Here are the formulae for triangles presented in a table format:

Property	Formula
Area of a Triangle	Area = 1/2 x Base x Height
Perimeter of a Triangle	Perimeter = a + b + c
Semi-Perimeter of a Triangle	S = (a + b + c)/2
Heron’s Formula	A = √[s(s – a)(s – b)(s – c)]
Pythagorean Theorem	Hypotenuse2 = Base2 + Perpendicular2
Law of Sines	a/Sin(A) = b/ Sin(B) = c/Sin(C)
Law of Cosines	c2 = a2 + b2 − 2ab ⋅ cos(C) a2 = b2 + c2 − 2bc ⋅ cos(A) b2 = a2 + c2 − 2ac ⋅ cos(B)

These formulae are commonly used in geometry to compute the characteristics of triangles.

Examples on Properties of Triangle

Example 1: The sides of a triangle are 6 cm, 7 cm, and 9 cm. Find the perimeter and semi perimeter of the triangle.

Solution:

Sides of triangle are a = 6 cm, b = 7 cm, and c = 9 cm.

To calculate the perimeter of a triangle, use the formula P = a + b + c,

P = 6 + 7 + 9 = 22 cm

Therefore, the perimeter of the supplied triangle is 22 cm.

To calculate the semi perimeter of a triangle, use the formula P = a( + b + c) / 2,

P = (6 + 7 + 9) / 2

P = 22/2 = 11 cm

Therefore, the semi perimeter of the supplied triangle is 11 cm.

Example 2: The measure of two angles in a triangle is 75^∘ and 85^∘. What will be the third angle’s measurement?

Solution:

Angles of a triangle are measured as 75^∘ and 85^∘.

Sum of two angle measurements 75^∘ + 85^∘ = 160^∘

Total of all three angles in a triangle = sum property equals 180 degrees.

Hence, the measure of the third angle = 180^∘−160^∘=20^∘.

Example 3: A triangle with sides of 6 cm, 8 cm, and 9 cm (with 8 cm as the base) has an altitude of 5.5 cm. Calculate the area of the triangle.

Solution:

Given:

• Base = 8 cm
• Height = 5.5 cm

Area of a Triangle(A) = 1/2 × b(base) × h(height)

A = (1/2) × 8 × 5.5

A = 22 cm²

Practice Questions on Properties of Triangle

P1: The measure of two angles in a triangle is 45^∘ and 55^∘. What will be the third angle’s measurement?

P2: Joe needs to make a triangle with sides of 5 cm, 8 cm, and 7 cm. Is it feasible to build the triangle?

P3: A triangle with sides of 5 cm, 6 cm, and 8 cm (with 6 cm as the base) has an altitude of 4.5 cm. Calculate the area and perimeter of the triangle.

P4: Find the perimeter of the triangle PQR with sides PQ = 5 cm, QR = 7 cm, and RP = 6 cm.

P5: If PQR with sides PQ = 5 cm, QR = 7 cm, and RP = 6 cm. Name the smallest and largest angles of a triangle.

FAQs on Properties of Triangle

What is a triangle, and how do you define it?

A triangle is a fundamental geometric form that has three sides and three angles. It arises when three points are joined by straight lines.

What are the different sorts of triangles, according to their sides and angles?

Triangles are classed as equilateral, isosceles, and scalene depending on their side lengths, and acute, right, and obtuse based on the angles.

What is the Angle Sum Property of a Triangle?

Angle Sum Property asserts that the sum of a triangle’s internal angles is always 180 degrees.

What is the importance of the Pythagoras Property in triangles?

Pythagoras Property, represented as c² = a² + b², applies to right triangles and helps determine the connection between the lengths of the sides.

Why is a triangle’s longest side opposite its biggest angle?

This characteristic is visible and helpful in determining the longest side of a triangle given its angles.

What does the Exterior Angle Property indicate?

Exterior Angle Property states that each exterior angle of a triangle is the sum of its two remote interior angles.

What is the Congruence Property of Triangles?

Triangles are congruent if they have both the same size and form. This characteristic is useful for detecting identical triangles using congruence criteria.

How are triangles utilized in real-world applications?

Triangles are commonly seen in everyday items such as signs, roofs, and bridges, and they serve as the foundation for many geometric ideas and computations.

What are the 6 types of triangles?

6 types of Triangles are:

1. Scalene Triangles
2. Isosceles Triangles
3. Equilateral Triangles
4. Acute Triangles
5. Right Triangles
6. Obtuse Triangles

What are the 5 properties of triangle Class 7?

Basic five properties of triangle are,

1. A triangle has three sides and three angles.
2. Sum of interior angles of a triangle is always 180 degrees.
3. Sum of exterior angles of a triangle is always 360 degrees.
4. Sum of consecutive interior and exterior angle is supplementary.
5. Sum of the lengths of any two sides of a triangle is greater than the length of the third side.