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A triangle is a closed geometrical shape with three sides, three vertices, and three angles. It is a polygon with the fewest sides. Triangles can be divided into two categories based on the sides and based on angles, like equilateral triangles, scalene triangles, isosceles triangles, etc. Triangles are used in many areas, such as building, art and design, locating objects, truss bridges, etc. In daily life, many things are triangular, like sandwiches, pizza, pyramids, etc.
These notes cover the basic definition of the triangle, triangle congruence, SAS, ASA, SSS and RHS Congruence Rule, and triangle Properties, triangle Inequalities. CBSE Class 9 chapter includes examples for the given topics as required. Students must practice all the questions and examples to ace the topic. NCERT class 9th maths notes contain all the chapters as per the latest syllabus. Students can download NCERT notes according to their standard and subjects.
NCERT Class 9 Maths Chapter 7 Triangles: Notes
Triangle
The word triangle comes from the Latin word triangulus, and the meaning of triangulus is three-cornered or having three angles. Therefore, the triangle is a closed figure that has three sides.
Congruent Triangles
If in a pair of triangles all three sides as well as three angles are equal, then this is called the congruence of triangles, and it is said Corresponding parts of the congruent triangles (CPCT), and it is denoted by the symbol $≅$. To prove the congruence of triangles, there are some criteria, and these criteria are discussed below.
Criteria for Congruence of Triangles
Triangle congruence is one of the important concepts not only in mathematics but also in real life. For example, congruency is used to create identical objects. For defining the congruency of triangles, there are some criteria, and these are as follows.
1. SAS
2. ASA
3. AAS
4. SSS
5. RHS
SAS Criteria for Congruency
If the two sides of the pair of triangles and the angle that is included by these two sides are equal to the corresponding sides and angle that includes these two sides, then this is called Side-Angle-Side congruence (SAS), as shown in the figure.
ASA Criteria for Congruency
If two angles of two triangles and the side that includes these two angles are equal to the corresponding two angles and the side that includes these angles is equal is called the Angle-Side-Angle congruence (ASA), as shown in the figure.
AAS Criteria for Congruency
If in two triangles, two angles and one side are equal to the two angles and one side of another triangle, then this is called the Angle-Angle-Side Congruence (AAS), as shown in the figure.
SSS Criteria for Congruency
If three sides of the two triangles are equal, then the two triangles then this is called the Side-Side-Side congruency(SSS), as shown in the figure.
RHS Criteria for Congruency
If in two right-angle triangles the right angle, hypotenuse and the other side are equal, then this is called the Right-Hypotenuse-Side congruency (RHS) as shown in the figure.
Important Note about SSA and AAA Congruency Rules
1. SSA is not a valid rule as it does not include the angle between the pair of sides
2, AAA is not a valid rule, as all three angles of both triangles may be equal, but the lengths of the sides are different.
As shown in the figure,
Properties of a Triangle
1. Angles opposite to the equal sides of the triangle are equal in an isosceles triangle.
2. If any two sides are unequal in a triangle, then the angle opposite to the longer side of the triangle will be greater than the angle opposite to the shorter side.
3. The sum of any two sides of the triangle is greater than the size of the third side.
Class 9 Chapter Wise Notes
Students must download the notes below for each chapter to ace the topics.
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NCERT Class 9 Solutions for Maths And Science
Students must check the NCERT solutions for Class 10 Maths and Science given below:
NCERT Class 9 Exemplar Solutions for Maths And Science
Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:
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