Payal Tandon
Co-founder, e-GMAT
Welcome to e-GMAT Support!
I am Payal, Co-Founder of e-GMAT.
Feel free to ask any Query.
Thank you for your query.
We will be contacting you soon on
A 8 min read

## Introduction

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:

Video Lesson on Triangles :

Here is an outline of the topics we will cover in this article:
[hide]

Questions on triangles are very commonly asked on the GMAT. Ace GMAT Quant by signing up for our free trial and get access to 400+ questions. We are the most reviewed online GMAT Prep company with 2700+ reviews on GMATClub.

## 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.

Did you know that a high GMAT score can increase your chances of admission to your dream B-school? Moreover, a GMAT score of 730+ yields an incremental \$500k in ROI. Here is how we can help:

Begin your GMAT preparation with the only prep company that has delivered more 700+ scores than any other GMAT club partner. Achieve GMAT 740+ with our AI driven tools that you personalized feedback at every step of your GMAT journey. Take our free trial today!

## 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 | Properties of 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.

Questions on triangles are very commonly asked on the GMAT. Ace GMAT Quant by signing up for our free trial and get access to 400+ questions. We are the most reviewed online GMAT Prep company with 2700+ reviews on GMATClub.

Save 60+ hours of GMAT preparation by crafting a well-defined study plan in just 3 steps:

### 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 | Properties of 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°

Questions on triangles are very commonly asked on the GMAT. Ace GMAT Quant by signing up for our free trial and get access to 400+ questions. We are the most reviewed online GMAT Prep company with 2500+ reviews on GMATClub.

## Special cases of Right Angle Triangles | Special properties of Triangle

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

## Properties of Triangle: Summary & Key Takeaways

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

Starting with your GMAT preparation? Here is a 5 step preparation plan to ace the GMAT:

Take a free GMAT mock to understand your baseline score and start your GMAT prep with our free trial. We are the most reviewed online GMAT Prep company with 2500+ reviews on GMATClub

## Practice Questions with Solution

### Question: 1

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

1. 20°
2. 40°
3. 60°
4. 80°
5. 100°

### Solution

Step 1: Given

• ∆DEF is an isosceles triangle
• ∠D = 100°

Step 2: To find

• The value of ∠F

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?

1. 120
2. 130
3. 240
4. 260
5. 312

### 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.

For more practice questions on triangles and geometry, sign up for our free trial. We have more than 400+ practice questions and 10+ hours of AI-driven video lessons.

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

Did you know e-GMATers have reported more 700+ scores than ever before in GMAT Club’s history? Watch this video to understand how e-GMAT has achieved this record-shattering result by investing and innovating with a single goal in mind – To create a platform that empowers students to achieve and deliver their very best.

If you are planning to take the GMAT, we can help you with a personalized study plan and give you access to quality online content to prepare. Write to us at acethegmat@e-gmat.com. We are the most reviewed GMAT prep company on gmatclub with more than 2700 reviews and are the only prep company that has delivered more than 700+ scores than any other GMAT club partner. Why don’t you take a free trial and judge for yourself?

## 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.