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Triangle Theorems

Last updated date: 27th Feb 2024
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About Triangle

Triangle is a plane figure consisting of three sides, and three angles opposite to the sides makes a sum of 180 degrees. It comes in the category of a polygon; polygon is a closed shape plane with line segments. Triangle is the simplest polygon in geometry.

Triangles theorems have significant use in geometry; they prove various properties associated with it. To give students in-depth knowledge about triangle theorems, Vedantu has uploaded the well-researched explanations on the website. Understanding the theorems is necessary to solve the problems linked with triangles.


Properties of a Triangle

The properties of a triangle include the followings:

  • It has three sides, angles, and vertices

  • The sum of three interior angles are always 180 degree

  • The sum of the two sides of this geometrical figure is greater than its third one

  • The area of the product of this figure’s height and the base is equal to twice its area.


Types of Triangle

There are different types of triangles, and here are the classifications –

According to the Measurement of Angles

  • Acute angle, where all interior angles are less than 90 degrees.

  • Right angle, where one of the three interior angles of a triangle is 90 degrees.

  • Obtuse angle, where one interior angle is greater than 90 degrees.


According to the Measurement of Sides

  • An equilateral triangle is where all 3 sides are equal.

  • An isosceles triangle is where 2 sides are equal.

  • A scalene triangle is where no sides are equal.

Since the definition of triangles and their types are now clear, students can now understand the theorems quicker.


Theorems of Triangle

Triangle theorems are based on various properties of this geometrical shape, here are some prominent theorems associated with this is that students must know –

1. Pythagoras Theorem

Probably the most popular and widely discussed triangle theorems are Pythagoras’ one.

Pythagoras theorem Class 10 states that ‘in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. According to this theorem, the sides of a triangle are named perpendicular, hypotenuse, and base. The hypotenuse is the longest one among these three sides, and it sits opposite to the right angle, i.e. 90 degrees.


Moreover, when sides of a Pythagoras theorem triangle have a positive integer value, and then squared and entered into an equation, they are known as a Pythagorean triple.

Formula of Pythagoras theorem: Hypotenuse2 = Perpendicular2 + Base2 


2. Triangle Similarity Theorems

The focus of this theorem is to prove the similarity between two triangles. It specifies conditions under which more than one triangle can be regarded as similar. It considers sides and angles to conclude, and once every condition is met, triangles are considered the same.


There are three subtypes of this triangle theorem. These are –

  • AA Similarity Theorem: 

This theorem suggests that if two angles of two triangles are similar, they have the same properties. Since the total value of three angles is 180 degrees, once the value of two angles is known, it is easy to find the third one by subtracting it from 180.

  • SSS Theorem: 

This theorem suggests that when 3 sides of two triangles have the same value or they are proportionate to each other, they are identical or congruent. Moreover, for these two, three sides need to be proportional.

  • SAS Theorem: 

When two sides of two triangles are proportional, and the angle between them have similar values, these two triangles will be similar.


3. Basic Proportionality Theorem

The Triangle proportionality theorem suggests that, when a line is drawn matching to one side of a triangle intersecting the other two at particular points, these other two sides are divided in the same ratio.


4. Triangle Sum Theorem

Probably the most basic among every triangle theorem, this one proves that all-three angles of this geometric figure constitute a total value of 180 degrees.


5. Triangle Inequality Theorem

The Triangle inequality theorem suggests that one side of a triangle must be shorter than the other two. Otherwise, they will not meet and create a triangle. These are some of the notable theorems associated with triangles. Students can learn more about them from the website of Vedantu – India’s leading e-learning platform.


At the website and mobile application of Vedantu, students will find relevant information and explanation regarding triangle theorems. Furthermore, there are various study materials to aid students in this regard. Also, they can sign up for online classes and doubt clearing sessions to better their preparations for this concept.

Learn Triangle Properties on Vedantu

Learn about the properties of the triangle explained by the subject matter expert of the Vedantu. Properties of a triangle help you understand different theorems and find a difference between the various types of them.

On Vedantu, you get a chance to learn from India’s best maths faculties. Join the online maths tuition class to clear your doubts from them and boost your preparation to score good marks in the exams.

Triangle Theorems Explanation

Theorems are a statement that has definite proof. For example, the Pythagoras theorem proves a square of the hypotenuse is equal to the sum of the squares of the other two sides.

Vedantu has explained the types and theorems of triangles to help students understand the shape that will help them solve geometrical problems effortlessly. These theorems cover the basics of a triangle that stays with you for a long duration. And benefits you in solving numerical in the higher classes and also in the engineering or architecture stream.

FAQs on Triangle Theorems

1. What is Triangle Inequality?

Inequalities in triangles represent three factors. Firstly, if two sides of a triangle are not similar, the angle on the opposite of the longer side is bigger than others. To prove this theory, one can take an example of a scalene triangle where no side is similar, and angle on the opposite to the longer side is the greatest one, and vice-versa. Secondly, the sum of two sides of a triangle is higher than the third one. This theory is very easy to prove; one can do it using any triangle.

2. How to Prove if Triangles are Similar?

To prove whether two triangles are similar or not, there are few theories. If all the sides are equal to two triangles, then they are identical. Also, if two sides and the angle between them are similar in two triangles, they are similar. Moreover, when two angles and their adjacent sides are similar in two figures, it proves their similarity. Finally, when two angles and a non-included side of two triangles are the same, it proves their similarity.

3. What is Congruence?

In Geometry, two figures are congruent when they have similar shapes and sizes. Moreover, two-point becomes congruent when one can transform into the other via an isometry. It includes rigid motions like translation, rotation, or reflection.

4. Why should you read triangle theorems?

Triangle is a simple polygon with three sides and angles, but when you face numerous triangles at once or their types, it becomes slightly complex to solve, and at the time, these theorems help you get the solution of the problems. Without theorems, triangle geometry is like a body without a soul, and more often, you will find an examiner asking these theorems in the examination for proof. Vedantu experts have explained all of these things in detail, and you can even download the problems based on triangle theorems from the website.

5. What are the benefits of using Vedantu to learn triangles?

Vedantu provides you with the best maths teachers; they break down most complex topics into simple points that are easy to swallow for the students. Subject matter experts at Vedantu use 3-D visuals to explain the topics and provide external resources to give you a better understanding.  You can find more materials linked to the triangles like past questions, articles, and formulas. To get the benefit of all these things, register to Vedantu pro classes with your email id or phone number.

6. How to study triangles to score more marks?

To score good marks in maths or any topic of it; only one thing that can help is practising a lot of numeral problems linked with topics. Vedantu has explained the triangle from the basics so that you do not have any doubts, and still, if you have any, you can clear it with the expert in one-on-one sessions. These sessions are for the students who have joined online maths tuition classes. 

Take a note of the theorems and types explained by the subject matter expert, and start solving new questions with Vedantu to ace your exam.

7. How to prove two triangles are similar?

To prove triangles are similar, you need to use the conditions explained in the triangle similarity theorem. This theorem has included three subtypes that prove triangles are similar. These conditions or subtypes consider the side and angles of the triangle, and triangles are similar only if conditions match. Subtypes of triangle theorems are the AA similarity theorem, SSS similarity theorem, and SAS theorem.

  • AA Similarity Theorem

If two angles of the triangles are similar, then they have the same properties.

  • SSS Similarity Theorem

It says when three sides of two triangles are the same or in the same ratio, then they are congruent or similar to each other.

  • SAS Similarity Theorem

It says when two sides of two triangles are proportional to each other, and the angle between them is similar or the same, then triangles are similar.

8. How do triangle theorems help to solve problems? 

Solving a numerical problem in mathematics requires thorough knowledge of the concept. You have to read and practice solved examples rigorously to reach the depth of the topics, and Vedantu makes this task easy for you with its online resources and tuition class. You get to learn each triangle theorem from the maths expert, that helps you with your doubts and provides other materials for preparations. These theorems help you solve questions based on similarity and inequality.