Triangles Notes
This chapter is about congruence of triangles — when two triangles are exact copies of each other. It covers the congruence criteria (SAS, ASA, AAS, SSS, RHS), properties of an isosceles triangle, and basic inequalities relating the sides and angles of a triangle.
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- 12 practice questions
- Aligned to CBSE 2025–26 syllabus
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Use this map for a first read, then return to the checks and formulas during revision.
- 1.Congruence and the SAS, ASA, AAS Criteria
- 2.SSS and RHS Criteria; CPCT
- 3.Properties of an Isosceles Triangle
- 4.Inequalities in a Triangle
- 2 key formulas
- 2 concept checks
Congruence and the SAS, ASA, AAS Criteria
Two figures are congruent if they have exactly the same shape and size. For triangles, congruence means the three sides and three angles of one equal the corresponding parts of the other. We write △ABC ≅ △PQR, keeping the letters in corresponding order.
SAS criterion: two triangles are congruent if two sides and the included angle of one equal the corresponding two sides and included angle of the other.
ASA criterion: congruent if two angles and the included side match. AAS is a form of this: if two angles and a non-included side match, the triangles are congruent (because the third angle is then also equal).
SSS and RHS Criteria; CPCT
SSS criterion: two triangles are congruent if the three sides of one are equal to the three sides of the other.
RHS criterion (for right triangles): two right triangles are congruent if the hypotenuse and one side of one equal the hypotenuse and one side of the other.
Once two triangles are shown congruent, all their corresponding parts are equal — abbreviated CPCT (Corresponding Parts of Congruent Triangles). This is how most further results (equal segments, equal angles, bisected lines) are proved.
Properties of an Isosceles Triangle
In an isosceles triangle the two equal sides are the legs and the third is the base. Angles opposite the equal sides are equal (the base angles). This is proved by drawing the bisector of the vertex angle and using SAS.
The converse is also true: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.
Consequently, in an equilateral triangle all three angles are equal (each 60°), and a triangle with all angles equal is equilateral.
Inequalities in a Triangle
In any triangle, the side opposite the greater angle is longer, and conversely the angle opposite the longer side is greater. So the longest side of a triangle faces its largest angle.
Triangle inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Equivalently, the difference of any two sides is less than the third side.
This is why lengths like 2 cm, 3 cm, 8 cm cannot form a triangle (2 + 3 < 8), while 4 cm, 5 cm, 7 cm can.
Triangles: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Congruence and the SAS, ASA, AAS Criteria
- SSS and RHS Criteria; CPCT
- Properties of an Isosceles Triangle
- Inequalities in a Triangle
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (2)
Properties of an Isosceles Triangle
Inequalities in a Triangle
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
What is the difference between congruent and similar triangles?
Congruent triangles have the same shape and the same size (all corresponding sides and angles equal). Similar triangles have the same shape but possibly different sizes (corresponding angles equal, corresponding sides in proportion). Class 9 focuses on congruence.
Why is 'SSA' not a congruence rule?
Two sides and a non-included angle do not fix a triangle uniquely — two different triangles can be drawn with the same SSA data (the 'ambiguous case'). Only SAS, ASA, AAS, SSS, and RHS guarantee congruence.
What does CPCT mean and when is it used?
CPCT stands for 'Corresponding Parts of Congruent Triangles (are equal)'. Once two triangles are proved congruent by a criterion, CPCT lets you conclude that any pair of corresponding sides or angles is equal.
How do you quickly check whether three given lengths can form a triangle?
Add the two shorter lengths. If their sum is greater than the longest length, a triangle is possible; if it is equal or smaller, it is not.