Class 9 — Mathematics (NCERT)
Triangles: Congruence, Properties and Inequalities
Class 9
- ✓Understand the concept of congruence of triangles.
- ✓Apply the SSS, SAS, ASA, AAS, and RHS congruence rules to prove triangles congruent.
- ✓State and apply the properties of isosceles triangles.
- ✓Understand and apply the theorems related to inequalities in a triangle.
- ✓Solve problems involving congruence, properties, and inequalities of triangles.
Key concepts
Two geometric figures are congruent if they are exactly identical in shape and size. For triangles, this means that if ΔABC is congruent to ΔPQR, then their corresponding sides are equal (AB=PQ, BC=QR, CA=RP) and their corresponding angles are equal (∠A=∠P, ∠B=∠Q, ∠C=∠R). We write this as ΔABC ≅ ΔPQR.
If three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.
If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
If in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.
Theorem 1: Angles opposite to equal sides of an isosceles triangle are equal. (If AB = AC, then ∠B = ∠C).\nTheorem 2: The sides opposite to equal angles of a triangle are equal. (If ∠B = ∠C, then AB = AC). This is the converse of Theorem 1.
Theorem 1: If two sides of a triangle are unequal, the angle opposite to the longer side is larger (greater).\nTheorem 2: In any triangle, the side opposite to the larger (greater) angle is longer. (Converse of Theorem 1).\nTheorem 3 (Triangle Inequality Theorem): The sum of any two sides of a triangle is greater than the third side. (e.g., AB + BC > AC, BC + CA > AB, CA + AB > BC).
Key facts to remember
- 1Two triangles are congruent if they have exactly the same shape and size.
- 2The five congruence rules for triangles are SSS, SAS, ASA, AAS, and RHS.
- 3CPCTC stands for 'Corresponding Parts of Congruent Triangles are Congruent'.
- 4In an isosceles triangle, angles opposite to equal sides are equal.
- 5In a triangle, sides opposite to equal angles are equal.
- 6The angle opposite the longer side in a triangle is larger.
- 7The side opposite the larger angle in a triangle is longer.
- 8The sum of any two sides of a triangle is always greater than the third side.
Worked examples
Example 1
In quadrilateral ACBD, AC = AD and AB bisects ∠A. Show that ΔABC ≅ ΔABD. What can you say about BC and BD?
Answer
ΔABC ≅ ΔABD, and BC = BD.
Example 2
In ΔABC, the bisector AD of ∠A is perpendicular to side BC. Show that AB = AC and ΔABC is an isosceles triangle.
Answer
AB = AC, and therefore ΔABC is an isosceles triangle.
Example 3
Show that in a right-angled triangle, the hypotenuse is the longest side.
Answer
The hypotenuse (AC) is the longest side.
This example demonstrates the relationship between angles and opposite sides.
Common mistakes
- ✗Assuming AAA (Angle-Angle-Angle) or SSA (Side-Side-Angle) are congruence rules. They are not.
- ✗Incorrectly identifying corresponding vertices, sides, or angles when writing congruence statements (e.g., writing ΔABC ≅ ΔPQR when it should be ΔABC ≅ ΔQPR).
- ✗Not distinguishing between included and non-included angles/sides for SAS/ASA/AAS rules.
- ✗Forgetting to state the congruence rule used in each step of a proof.
- ✗Applying the triangle inequality theorem incorrectly, especially when checking if three given lengths can form a triangle.
Exam tips
- ★Always draw a clear and labelled diagram based on the problem statement.
- ★Clearly list 'Given' information and what is 'To Prove' before starting the proof.
- ★In proofs, provide a reason for every statement (e.g., Given, Common side, Vertically opposite angles, By SAS congruence rule, By CPCTC).
- ★Ensure that the order of vertices in a congruence statement (e.g., ΔABC ≅ ΔPQR) correctly reflects the correspondence of parts.
- ★Practice a variety of problems to become proficient in identifying which congruence rule or inequality theorem to apply.
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