Class 10 Maths Chapter 6 Triangles Study Material In PDF
We already know a lot about triangles from previous classes, and in-class 10 maths chapter 6 triangles we will learn different concepts like the principle of similarity of figures, etc. It’s easy to measure small shapes and sizes using the measurement tape. e.g if someone asked you to measure the table in your house of school, then you can easily do it by using a measuring tape. But have you ever wondered how do they measure dimensions of exponentially big things like mountains?
To measure the dimensions of such things they introduced the concept of similarity of figures. With this concept, one can calculate things like distance from the earth to the moon, etc. Calculating astronomical values is a far concept, but in the triangles chapter, we will take a step towards it.
Class 10 Maths Chapter 6 Triangles
Before we proceed to learn the concept and topics of class 10 maths chapter 6 triangles, we first need to get familiar with important terms that will be used throughout this chapter. Here are some of those terms:
Different types of triangles:
- Equilateral triangle: The triangle in which all the sides are equal and each angle is 60° is called an equilateral triangle.
- Scalene triangle: The triangle in which all sides are different is called a scalene triangle.
- Isosceles triangle: The triangle having two equal sides is called the isosceles triangle.
- Right angle triangle: The triangle in which one angle is 90° is called the right angle triangle.
- Obtuse-angled triangle – The triangle in which one of the angles is greater than 90° is called an obtuse-angled triangle.
- Acute-angled triangle – The triangle in which all the angles are less than 90° is called the acute-angled triangle.
What is a polygon?
The word polygon is a combination of ‘poly’ meaning ‘many’ and ‘gon’ meaning ‘sides’. Following are the features of a polygon:
- It is a closed 2D shape that is made up of line segments and not curves.
- A polygon can have n numbers of line segments connected with each other.
- Minimum 3 line segments are required to form a polygon. This means that a triangle is also a polygon.
- A circle can not be called a polygon as it is made up of curves.
What is meant by congruence?
The plain English meaning of the term ‘congruence’ is compatibility or harmony. In Maths, when two shapes are the same in terms of shape and size, then they are said to be congruent. In simple words, if two shapes cover each other completely when placed over each other, then they are said to be congruent.
So, in terms of triangles, when two triangles are the same in shape and size, then they will be called congruent triangles. The sign to show congruency between two triangles is ≅.
What is CPCT?
When the corresponding parts of congruent triangles are equal then we write it as CPCT (corresponding parts of congruent triangles).
Class 10 Maths Chapter 6 Triangles – Previous Class Summary
All major general terminologies that will be used in this chapter are now out of the way, we can now move on to the summary of the previous class to refresh your memory.
- In a triangle: the angle opposite to the longer side is greater, the side opposite to the greater angle is longer, and the sum of any two sides is greater than the third side.
- Two circles are congruent if their radii are the same, and two squares are congruent if their sides are the same.
- The symbolic representation of two congruent triangles ABC and PQR under the correspondence A↔P, B↔Q, C↔R is given by ΔABC≅ΔPQR.
- Two triangles are said to be congruent if they satisfy one of the following congruence rules:
- SAS (Side-Angle-Side): If two sides and the included angle of two triangles are the same then they are congruent by SAS rule.
- ASA (Angle-Side-Angle): Two angles and the included side of two triangles are the same then they are congruent by ASA rule.
- AAS (Angle-Angle-Side): when two angles and one side of a triangle is equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent by the AAS rule.
- SSS (Side-Side-Side): If three sides of a triangle are equal to three sides of another triangle then the triangles are congruent by the SSS rule.
- RHS: In two right triangles, if the hypotenuse and one side of a triangle are equal to the hypotenuse and one side of the other triangle, then they are congruent by the RHS rule.
- Angles opposite to equal sides of a triangle are equal, and sides opposite to equal angles are equal.
Class 10 Maths Chapter 6 Triangles – Explanation
After revisiting the topics of triangles from the previous topic, let’s take a quick look at all the topics that we will learn in class 10 maths chapter 6 triangles.
# Similar figures
when two figures have identical shapes but differ in the size, then they are called similar figures. So, we can say that all the congruent figures are similar but not all similar figures are congruent.
# Similarity of triangles
Two polygons with the same number of sides are similar only if their corresponding angles
are equal, and their corresponding sides are in the same proportion. If we draw a line parallel to one side of a triangle such that it intersects the other two sides at different points, then the other two sides are divided in the same ratio. And that line will be parallel to the third side.
# Criteria for similarity of triangles
In order for two triangles to be similar they must satisfy one of the following criteria:
- AAA (Angle-Angle-Angle): When corresponding angles of two triangles are equal, then their corresponding sides will be in the same ratio and hence the two triangles will be similar by the AAA similarity criterion.
- AA (Angle-Angle): When two angles of one triangle are respectively equal to the two angles of
the other triangle, then the triangles are similar by AA similarity criterion.
- SSS (Side-Side-Side): When corresponding sides of two triangles are in the same ratio, then it means that their corresponding angles are also equal which makes them similar by SSS similarity criterion.
- SAS (Side-Angle- Side): If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar by SAS similarity criterion.
# Areas of Similar Triangles
The area of two similar triangles can be found by squaring the ratio of their sides.
# Pythagoras Theorem
We have studied the Pythagoras theorem in previous classes, in class 10 maths chapter 6 triangles we will prove this theorem. The theorem will lead us to the following conclusions.
If we draw a perpendicular from the vertex of the right angle of a right triangle to its hypotenuse, then the small triangles that will form on both sides of the perpendicular will be similar to each other and also to the whole triangle.