Triangle Theorems

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A triangle is a polygon with three sides, three vertices, and three edges. An angle is formed when two sides meet, and a triangle contains three such angles where their total sum is 180 degree. It is one of the most fundamental figures of geometry, and it is a two dimensional one.

Triangle theorems are associated with various subtypes of this geometric figure, and they prove various properties associated with it. Having an in-depth knowledge of such theorems is essential to understand a triangle thoroughly and multiple problems associated with them.

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 two sides of this geometrical figure is greater than its third one

  • Area of the product of this figure’s height and the base is equal to twice of 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 degree.

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

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 its 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 among triangle theorems is 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 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 as 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

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 theorems, this one proves that all-three angles of this geometric figure constitute a total value of 180 degrees.

5. Triangle Inequality Theorem

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.

FAQ (Frequently Asked Questions)

1. What is Triangle Inequality?

Ans. 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?

Ans. 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?

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