Triangles: Similarity & Pythagoras Theorem

A triangle can be stated as a polygon, which has three sides and three angles. The interior angle of any triangle sums up to 180 degrees. Whereas an exterior angle of a triangle makes a total of 360 degrees. In this article, we will look at the various aspects of the triangle. We will cover the classification, similarity, and Pythagoras theorem related to triangles.

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Types of Triangles

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If we consider triangle’s lengths and angles, a triangle can be classified into multiple categories, like,

1. Scalene Triangle- When all three sides of the triangle are of different measure, it is said to be a Scalene Triangle.
2. Isosceles Triangle- When two sides of a triangle are of the same length, it is stated as an isosceles Triangle
3. Equilateral Triangle- When all three sides of a triangle are of equal length, and each angle of a triangle is 60 degrees, it is said to be an equilateral triangle.
4. Acute angled Triangle- When all angles of a triangle are less than 90 degrees, it is said to be an Acute Angled Triangle.
5. Right Angle Triangle- When any one of the angles of a triangle is equal to 90 degrees, it is said to be a Right Angle Triangle.
6. Obtuse Angle Triangle- When any of the angles of a triangle is greater than 90 degrees, it is said to be an obtuse angle triangle.

The video below explains this:

Pythagoras Theorem Detailed Video Explanation:

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Similarity of Triangle

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The similarity of a triangle is determined by two statements. Two triangles are said to be the same when they fulfill two properties.

• their corresponding angles are equal.
• their corresponding sides are in the same proportion/ratio.

In the above diagram, Triangle KMN is similar to Triangle KPQ.

Equilateral triangle

According to the Mathematician Thales, two triangles are said to be equilateral triangles if the corresponding angles of two triangles are equal. Adding to that, two triangles should also match the next property given by Thales, the ratio of any of the two corresponding sides of two equiangular triangles is always the same.

Equilateral triangle

In the above diagram, Triangle ABC is equilateral to triangle BDE.

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Criteria for similarity of Triangle

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To prove the similarity of the triangle, you should look for the given criteria.

• Side-Side-Side (SSS) Similarity Criterion – Two triangles are said to be similar when two corresponding sides of two triangles are in the same ratio. And if their corresponding sides are equal, their corresponding angle will also be equal. By this, two triangles can be states as similar.
• Side-Angle-Side (SAS) Similarity Criterion – Two triangles are said to be similar if the angle of one triangle is equal to that of another triangle and the sides that include these angles are in the same ratio or proportional. Hence, two triangles will be proved to be similar.
• Angle Angle Angle (AAA) Similarity Criterion – Two triangles are said to be the same when the corresponding angle of two triangles are equal. By this, their corresponding side will also be in the same ratio. Hence, two triangles will be proved the same.
• Angle-Angle (AA) Similarity Criterion – Two triangles will be proved the same when two angles of one triangle are respectively equal to the two angles of the other triangle.

By using the above four principles, you can determine the similarity of the two triangles.

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Pythagoras Theorem

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According to Pythagoras Theorem, “In a right-angled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”

or

The diagonal of a rectangle produces by itself the same area as produced by both sides (i.e., length and breadth).

Conditions: To prove the Pythagorean theorem two conditions should be fulfilled.

1. It should be a right-angled triangle.
2. The sum of the square of the two sides should be equal to the square of the third side.

Explanation: The above triangle is a right-angle triangle as it has one angle equal to 90 degrees. So, if we add the square of the A-side and B sides, it will be equal to the square of the C side.