Properties of Triangles and Their Key Formulas

Any three-sided polygon having three edges and three vertices is referred to as a triangle in geometry.

The fact that a triangle's interior angles add up to \[180^\circ \] is its most crucial characteristic. Certain fundamental ideas, including the Pythagorean Theorem and trigonometry, depend on the characteristics of triangles.

What are the Properties of a Triangle?

We must understand the many sorts of triangles in order to learn about the properties of triangles. Although all triangles have some characteristics, some of these characteristics depend on the sides and angles of the triangle.

Angle Sum Property:

The angle sum property states that the sum of a triangle's three interior angles is always \[180^\circ \].

Angle Sum Property

In the given triangle, \[\angle A + \angle B + \angle C = 180^\circ \] 

Triangle Inequality Property

The length of a triangle's two sides added together is longer than its third side, according to the triangle inequality theorem.

Inequality Property

Pythagoras Theorem:

The hypotenuse square of a right-angled triangle is equal to the sum of the squares of the other two sides, according to the Pythagoras theorem. Mathematically, it is expressed as \[Hypotenus{e^2} = Bas{e^2} + Altitud{e^2}\]. See the altitude, base, and hypotenuse in the illustration below.

Side Opposite to the Greater Angle is the Longest Side:

Look at the triangle below to better grasp this triangle's property that the longest side is the one that is opposite the largest angle. B is the largest angle in this triangle. The side AC is hence the longest side.

Exterior Angle Property:

The exterior angle of a triangle is always equal to the sum of the interior opposite angles, according to the outside angle theorem. Exterior angle (e) of the triangle presented is equal to \[\angle a + \angle b\]

Exterior Angle Property

It should be remembered that a triangle can expand its three external angles, and the sum of all these exterior angles is \[360^\circ \].

Important Notes for Triangle:

• What is the formula of triangle area? The fundamental formula for determining a triangle's area is \[{\rm{Area of triangle  =  }}\dfrac{1}{2} \times Base \times Height\].

• A triangle's perimeter is the sum of the lengths of its three sides.

Conclusion:

To learn about the properties of triangles, we must first understand the many types of triangles. Although all triangles have some characteristics, some of these features are dependent on the triangle's sides and angles.

Solved Example:

Example 1: A triangle has two angles that are \[75^\circ \] and \[60^\circ \] in length. Determine the third angle's measurement.

Solution: Two angles in a triangle have measurements of  an\[75^\circ \]and \[60^\circ \].

Sum of two angles is \[135^\circ  = 75^\circ  + 60^\circ \].

Using a triangle's characteristics, we can determine that the total of its three angles equals 180°.

So the third angle will be $180^\circ -135^\circ =45^\circ $.

Example 2: Tim is trying to build a triangle with sides that are 5 cm, 4 cm, and 9 cm long. Can he accomplish it?

Solution: The sides are 5 cm, 4 cm, and 9 cm long.

5 cm + 4 cm= 9 cm

In this case, the third side is equal to the total of the two smaller sides. However, according to the triangle inequality theorem, any two sides should add up to more than the third side.

Tim won't be able to build a triangle with sides of 5 cm, 4 cm, and 9 cm, according to the triangle's characteristics.

Example 3:  The sides of a triangle are given as 3 cm, 4 cm, and 5 cm. Calculate the perimeter of the triangle.

Solution: Sides of the triangle are: x = 3 cm, y = 4 cm and z = 5 cm

The perimeter of the triangle is given by 

\[\begin{array}{l}P = x + y + z\\P = 3 + 4 + 5\\P = 12cm\end{array}\] 

Therefore, the perimeter of the given triangle is 12 cm.