What is true about the angles of a triangle?

In this article, we are going to learn about the simplest form of a polygon, a triangle. All polygons can be divided into triangles, or in other words, they are formed by combining two or more triangles. Thus, understanding the basic properties of a triangle and its types is essential.

There are six types of triangles in total – Isosceles, Scalene, Equilaterial, Oblique, Acute, and Right. Based on the classification according to internal angles, there are three types – Equilateral, Isosceles, and Scalene. Whereas, the types of a triangle that are classified according to the length of its side are Right, Acute, and Oblique. Here are the types of triangles:

Watch this video to know the basic property of triangle:

What is a triangle?

As the name suggests, the triangle is a polygon that has three angles. So, when does a closed figure has three angles?

When it has three line segments joined end to end.

Thus, we can say that a triangle is a polygon, which has three sides, three angles, three vertices and the sum of all three angles of any triangle equals 180°.

Properties of a triangle

The properties of a triangle are:

1. A triangle has three sides, three angles, and three vertices.
2. The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.
3. The sum of the length of any two sides of a triangle is greater than the length of the third side.
4. The side opposite to the largest angle of a triangle is the largest side.
5. Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Types of triangles

Triangles can be classified in 2 major ways:

• Classification according to internal angles (Right, Acute, Oblique)
• Classification according to the length of its sides (Equilateral, Isosceles, Scalene)

Let’s look into the six types of triangles in detail:

1. Acute Angled Triangle
2. Right-Angled Triangle
3. Oblique Angled Triangle
4. Scalene Angled Triangle
5. Isosceles Angled Triangle
6. Equilateral Angled triangle

Acute Angle Triangle

A triangle that has all three angles less than 90° is an acute angle triangle.

• So, all the angles of an acute angle triangle are called acute angles

Given below is an example of an acute angle triangle.

Right-Angle Triangle

A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• The other two angles of a right-angle triangle are acute angles.
• The side opposite to the right angle is the largest side of the triangle and is called the hypotenuse.

In a right-angled triangle, the sum of squares of the perpendicular sides is equal to the square of the hypotenuse.

For e.g. considering the above right-angled triangle ACB, we can say:

(AC)^2 + (CB)^2 = (AB)^2

This is known as Pythagoras theorem

Vice versa, we can say that if a triangle satisfies the Pythagoras condition, then it is a right-angled triangle.

Obtuse/Oblique Angle Triangle

A triangle that has one angle that measures more than 90° is an obtuse angle triangle.

Given below is an example of an obtuse/oblique angle triangle.

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Scalene triangle

A triangle that has all three sides of different lengths is a scalene triangle.

• Since all the three sides are of different lengths, the three angles will also be different.

Given below is an example of a scalene triangle

Isosceles triangle

A triangle that has two sides of the same length and the third side of a different length is an isosceles triangle.

• The angles opposite the equal sides measure the same.

Given below is an example of an isosceles triangle.

Equilateral triangle

A triangle that has all three sides of the same length is an equilateral triangle.

• Since all the three sides are of the same length, all the three angles will also be equal.
• Each interior angle of an equilateral triangle = 60°

Special cases of Right Angle Triangles

Let’s also see a few special cases of a right-angled triangle

45-45-90 triangle

In this triangle,

• Two angles measure 45°, and the third angle is a right angle.
• The sides of this triangle will be in the ratio – 1: 1: √2 respectively.
• This is also called an isosceles right-angled triangle since two angles are equal.

30-60-90 triangle

In this triangle,

• This is a right-angled triangle, since one angle = 90°
• The angles of this triangle are in the ratio – 1: 2: 3, and
• The sides opposite to these angles will be in the ratio – 1: √3: 2 respectively
• This is a scalene right-angled triangle since all three angles are different.

Area of Triangle

• Area of any triangle = ½ * base * height
• Area of a right-angled triangle = ½ * product of the two perpendicular sides

Let us summarize some of the important properties of a triangle.

• The sum of all interior angles of any triangle is equal to 180°
• The sum of all exterior angles of any triangle is equal to 360°
• An exterior angle of a triangle is equal to the sum of its two interior opposite angles
• The sum of the lengths of any two sides of a triangle is always greater than the length of the third side
• Similarly, the difference between the lengths of any two sides of a triangle is always less than the length of the third side
• The side opposite to the smallest interior angle is the shortest side and vice versa.
• Similarly, the side opposite to the largest interior angle is the longest side and vice versa.
  • In the case of a right-angled triangle, this side is called the hypotenuse
• The height of a triangle is equal to the length of the perpendicular dropped from a vertex to its opposite side, and this side is considered the base

If you liked this article, you may also like to read the following advanced level articles on triangles

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Properties of Triangle: Practice Question

Question: 1

In an isosceles triangle DEF, if an interior angle ∠D = 100° then what is the value of ∠F?

Solution

Step 1: Given

• ∆DEF is an isosceles triangle

Step 2: To find

Step 3: Approach and Working out

• We know that the sum of all interior angles in a triangle = 180°
• Implies, ∠D + ∠E + ∠F = 180°
• ∠E + ∠F = 1800 – 1000 = 80°
• Since ∆DEF is an isosceles triangle; two of its angles must be equal.
• And the only possibility is ∠E = ∠F
• Therefore, 2∠F = 80°
• Implies, ∠F = 40°

Hence the correct answer is Option B.

Question 2

In a right-angled triangle, ∆ABC, BC = 26 units and AB = 10 units. If BC is the longest side of the triangle, then what is the area of ∆ABC?

Solution

Step 1: Given

• ∆ABC is a right-angled triangle
  • BC = 26 units
  • AB = 10 units
  • BC is the longest side of the triangle

Step 2: To find

• The area of triangle ∆ABC

Step 3: Approach and Working out

• We are given that BC is the longest side of the triangle, which implies that BC is the hypotenuse

Thus, according to Pythagoras rule:

• BC2 = AB2 + AC2
• 262 = 102 + AC2
• AC2 = 676 – 100 = 576
• Therefore, AC = 24 units
• We know that the area of a right-angled triangle = ½ * product of the two perpendicular sides = ½ * AB * AC = ½ * 10 * 24 = 120 sq. units

Hence the correct answer is Option A.

Here are a few more articles that you may like to read:

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FAQ – Properties of a triangle

What is a triangle and its properties?

A triangle is a closed figure with three sides, three vertices, three angles, and the sum of internal angles is 180°

What are the different types of triangles?

Triangles can be classified in 2 ways, according to internal angles and according to the length of the sides. According to internal angles, there are three types of triangles i.e., acute, right, and obtuse-angled triangle. According to the length of sides, triangles can be classified into 3 categories i.e., Scalene, Isosceles, and Equilateral triangle.

What is a Scalene triangle?

A triangle that has all three sides of different lengths is a scalene triangle.

What is an Isosceles triangle?

A triangle that has two sides of the same length and the third side of a different length is an isosceles triangle.

What is an equilateral triangle?

A triangle that has all three sides of the same length is an equilateral triangle.