RD Sharma Chapter 9 Class 9 Maths Exercise 9.1 Solutions is based on the Triangle and its Angle. This chapter helps learn about the different angles of triangles like- Scalene Triangle, Acute Triangle, Right Triangle, Isosceles Triangle, etc. This exercise of Chapter 9 also includes the Angle sum property of a triangle. Here we will explain the triangles with a complete stepwise explanation and examples.
Moreover, for practicing more, we have attached the free PDF for the students in which several questions are mentioned with the easy solutions. With the help of PDF, learners will get to know the variety of questions of Triangles and its angle. So, download RD Sharma Chapter 9 Class 9 Maths Exercise 9.1 Solutions PDF for free and start practicing to score well in the exam.
Download RD Sharma Chapter 9 Class 9 Maths Exercise 9.1 Solutions PDF
Important Definitions RD Sharma Chapter 9 Class 9 Maths Exercise 9.1 Solutions
List of the topics and subtopics of Triangles and their angles-
- Types of triangles
- Scalene triangle
- Isosceles triangle
- Equilateral triangle
- Acute Triangle
- Right Angle Triangle
- Obtuse Triangle
- Angle sum property of a triangle
Types of Triangles
Triangles are classified into two types-
- On the basis of lengths of their sides (Scalene, Isosceles, Equilateral)
- On the basis of their interior angles (Obtuse, Acute, Right)
Triangles On the Basis of lengths of their Sides
The examples and definitions of the triangles based on the lengths of their sides are mentioned below in the following points.
A scalene triangle has each side length of different measures. No side is equal in length to any of the other sides. Every interior angle is also different in the scalene triangle.
In an Isosceles Triangle, the measures (length) of two of the three sides are equal. So, the angles opposite the corresponding sides are equivalent to each other. In simple words, it has two equal sides and two equal angles.
In an equilateral triangle, the lengths of the sides are equivalent. Each of the interior angles has a length of 60 degrees. Considering the angles of an equilateral triangle are the same, it is also called an equiangular triangle.
Triangles On the Basis of Their Interior Angles
Please look at the example and definitions of the triangle based on their Interior Angles in the following points.
Obtuse Angled Triangle
In Obtuse triangles, one of the three interior angles has a length greater than 90 degrees. In simple words, if one of the angles in a triangle is an obtuse angle, so the triangle is known as an obtuse-angled triangle.
Acute Angled Triangle
A triangle whose all three interior angles are acute is known as an Acute Triangle. In simple words, if each interior angle is less than 90 degrees, so it is called an acute-angled triangle.
Right Angled Triangle
A right triangle has one angle of 90 degrees. In a right-angled triangle, the side opposite to the right angle is the longest side known as the hypotenuse. As we have come across types of triangles with combined names like- a right isosceles triangle and so on, but this only signifies that the triangle has two equal sides with one of the interior angles remaining 90 degrees.
Angle Sum Property of a Triangle
An Angle sum property of a triangle signifies that the total of interior angles is 180°.
Prove An Angle Sum Property of a Triangle
Theorem 1- Prove that sum of each three-angle is 180° or 2 right angles.
To prove: ∠P + ∠Q + ∠R = 180°
Draw AB || QR passes through point P.
Proof: ∠1 = ∠Q and ∠3 = ∠R …. (i)
[ alternate angles ∵ AB || QR]
= APB is a line
= ∠1 + ∠2 + ∠3 = 180° (linear pair application)
∠Q + ∠2 + ∠R = 180°
∠Q + ∠RPQ + ∠R = 180°
= 2 right angles (Hence Proved).
Theorem 2- If one side of a triangle is given, then the exterior angle so set is equivalent to the sum of two interior opposite angles.
Means ∠4 = ∠1 + ∠2
Given, Means ∠4 = ∠1 + ∠2
= ∠3 = 180° – (∠1 + ∠2) …(1) (by an angle sum property) where, PQR is a line
= ∠3 + ∠4 = 180° (linear pair)
= or ∠3 = 180° – ∠4 …(2)
=by (1) & (2)
= 180° – (∠1 + ∠2) = 180° – ∠4
= ∠1 + ∠2 = ∠4 (Hence Proved).