# Triangles Class 10 Notes: Chapter 6

The triangles class 10 notes provided here are one of the most crucial study resources for the students studying in class 10. These CBSE chapter 6 notes are concise and cover all the concepts from this chapter from which questions might be included in the board exam. The main concepts from this chapter that are covered here are-

• What is a triangle?
• Similarity criteria of two polygons having the same number of sides
• Similarity criteria of triangles
• Proof of Pythagoras Theorem
• Example Questions
• Practice Questions
• Articles Related to Triangles

## What is a Triangle?

A triangle can be defined as a polygon which has three sides and three angles. The interior angles of a triangle sum up to 180 degrees and the exterior angles sum up to 360 degrees. Depending upon the angle and its length, a triangle can be categorized in the following types-

1. Scalene Triangle – three edges are of different length
2. Isosceles Triangle – has two equal sides
3. Equilateral Triangle – has three equal sides and has equal interior degrees of 60
4. Acute angled Triangle – has all the angles less than 90 degrees
5. Right angle Triangle – has one 90 degrees angle
6. Obtuse-angled Triangle – has an angle which is more than 90 degrees

### Similarity Criteria of Two Polygons Having the Same Number of Sides

Any two polygons which have the same number of sides are similar if the following two criteria are met-

1. Their corresponding angles are equal, and
2. Their corresponding sides are in the same ratio (or proportion)

### Similarity Criteria of  Triangles

There are four main criteria which determine whether two triangles are similar or not. These 4 criteria are:

• AAA Similarity Criterion – if the corresponding angles of any two triangles are equal, then their corresponding side will be in the same ratio and the triangles will be similar.
• AA Similarity Criterios – if two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.
• SSS Similarity Criterion – if the corresponding sides of any two triangles are in the same ratio, then their corresponding angles will be equal and they will be similar.
• SAS Similarity Criterion – if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar.

### Proof of Pythagoras Theorem

Statement – “In a right-angled triangle the sum of squares of two sides is equal to the square of the hypotenuse of the triangle.” Know more Pythagoras theorem and Pythagorean triplets here along with examples.

Proof –

Consider a right triangle, right angled at B.

Construction-

Draw $BD \perp AC$

Now, $\bigtriangleup ADC \sim \bigtriangleup ABC$

So, $\frac{AD}{AB} = \frac{AB}{AC}$

or, $AD. AC = AB^{2}$ ……………(i)

Also, $\bigtriangleup BCDC \sim \bigtriangleup ABC$

So, $\frac{CD}{BC} = \frac{BC}{AC}$

or, $CD. AC = BC^{2}$ ……………(ii)

$AD. AC + CD. AC = AB^{2} + BC^{2}$ $AC(AD + DC) = AB^{2} + BC^{2}$ $AC(AC) = AB^{2} + BC^{2}$ $\Rightarrow AC^{2} = AB^{2} + BC^{2}$

### Practice Questions

1. A girl having a height of 90 cm is walking away from a lamp-post’s base at a speed of 1.2 m/s. Calculate the length of that girl’s shadow after 4 seconds if the lamp is 3.6 m above the ground.
2. S and T are points on sides PR and QR of triangle PQR such that angle P = angle RTS. Now, prove that triangle RPQ and triangle RTS are similar.
3. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD
at F. Show that triangles ABE and CFB are similar.

### Articles Related to Triangles

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#### Practise This Question

What are the possibilities of outcomes when 2 coins are tossed in the air?