Triangles
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revise the CBSE Class 9 Triangles chapter: congruence of triangles through SSS, SAS, ASA, AAS and RHS rules, isosceles triangle properties, and triangle inequalities. Includes clear definitions, formulas, and practice questions with solutions.
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Congruent figures are exact copies of each other in shape and size. For triangles, verifying that all three sides and three angles match takes time, so the chapter introduces airtight shortcuts called criteria of congruence. SSS, SAS, ASA, AAS and RHS each state that only three correctly chosen corresponding parts are enough. Once congruence is established, every remaining side and angle becomes known through CPCT. This idea of proving equality of parts through congruent triangles is the main tool used throughout the chapter and in later geometry.
The criteria also let us study special triangles. By drawing an angle bisector or an altitude from the vertex of an isosceles triangle, the original triangle is split into two smaller triangles. Using SSS, SAS or RHS these smaller triangles can be proved congruent, which immediately gives the equal base angles of the isosceles triangle. Conversely, equal base angles force the two opposite sides to be equal. Thus the same congruence arguments yield the two-way property: equal sides imply equal base angles and equal base angles imply equal sides.
Towards the end, the chapter shifts from equality to comparison within a single triangle. The fundamental rule is that the larger angle is always opposite the longer side, and the smaller angle faces the shorter side. From this we get the triangle inequality: the sum of any two sides of a triangle must exceed the third side. This rule acts as a filter – if three given lengths fail it even once, no triangle can be constructed with them. It also lets us determine which side and angle in a triangle are the largest or smallest.
Key terms
- Congruent triangles
- Triangles having the same shape and the same size. One can be placed over the other so that their boundaries match; every side and angle has an equal corresponding part. The symbol ≅ is used, for example ΔABC ≅ ΔPQR.
- SSS criterion
- Side-Side-Side rule. Two triangles are congruent if all three sides of one are respectively equal to the three sides of the other. No knowledge of the angles is required.
- SAS criterion
- Side-Angle-Side rule. Two triangles are congruent if two sides and the angle included between them in one triangle are equal to the corresponding two sides and included angle of the other. The equal angle must be the one between the two given sides.
- ASA criterion
- Angle-Side-Angle rule. Two triangles are congruent if two angles and the side included between them in one triangle are equal to the corresponding angles and included side of the other.
- AAS criterion
- Angle-Angle-Side rule. Two triangles are congruent if two angles and a side opposite one of them in one triangle are equal to the corresponding two angles and side of the other. It is derived from ASA using the angle sum property.
- RHS criterion
- Right angle-Hypotenuse-Side rule. For right-angled triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another, then the triangles are congruent.
- Isosceles triangle property
- In an isosceles triangle, the angles opposite the equal sides are equal. The converse also holds: if two angles of a triangle are equal, then the sides opposite them are equal. This result is used to find missing angles and to prove side equalities.
- Angle–side inequality
- In any triangle, the larger angle faces the longer side and the smaller angle faces the shorter side. Therefore the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
- Triangle inequality
- For any triangle, the sum of the lengths of any two sides is greater than the length of the third side. Consequently, the difference between the lengths of any two sides is less than the third side. If three lengths fail this condition, they cannot be the sides of a triangle.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Triangle inequality (sum) | a + b > c, b + c > a, c + a > b | a, b and c are the lengths of the three sides of a triangle. The sum of any two sides must be strictly greater than the remaining side. |
| Triangle inequality (difference) | |a − b| < c | a, b and c are side lengths of a triangle. The difference between any two sides is always less than the third side; the expression shows this for a and b compared with c. |
| Angle–side ordering | ∠A > ∠B ⇔ BC > CA | In triangle ABC, BC is opposite ∠A and CA is opposite ∠B. A larger angle has a longer opposite side, and a longer side faces a larger angle. |
Practice questions with answers
1. What does it mean for two triangles to be congruent?
Congruent triangles are triangles that have exactly the same size and shape. When placed one over the other, their corresponding vertices, sides and angles coincide.
2. If ΔABC ≅ ΔDEF, write all the corresponding equal parts.
From the congruence statement, A matches D, B matches E, and C matches F. Hence AB = DE, BC = EF, CA = FD, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F.
3. Which congruence rule applies when the three sides of one triangle are equal to the three sides of another triangle?
The SSS (side-side-side) criterion applies, because three sides of one triangle are respectively equal to three sides of the other. No angle measurement is needed.
4. In ΔXYZ and ΔPQR, XY = PQ, ∠X = ∠P and XZ = PR. Which congruence rule proves that the triangles are congruent?
SAS criterion. The equal angle ∠X is included between the equal sides XY and XZ, just as ∠P is included between PQ and PR. Therefore the two triangles are congruent by SAS.
5. In triangles ABC and MNO, ∠A = ∠M, ∠B = ∠N and AC = MO. Name the congruence rule that makes them congruent.
AAS criterion. Two angles in each triangle are equal, and the side AC is opposite ∠B while MO is opposite ∠N. This is two angles and a non-included side, so AAS proves the triangles congruent.
6. State the RHS congruence criterion.
RHS criterion says that two right-angled triangles are congruent if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of the other. The right angle itself is already fixed in each triangle.
7. In an isosceles triangle ABC with AB = AC, if ∠A = 50°, what are the measures of ∠B and ∠C?
Since AB = AC, the base angles are equal: ∠B = ∠C. Let x = ∠B = ∠C. Using the angle sum property, 50° + x + x = 180°, so 2x = 130° and x = 65°.
8. Do the side lengths 2 cm, 3 cm and 6 cm form a triangle?
No. The triangle inequality requires the sum of any two sides to be strictly greater than the third. Here 2 cm + 3 cm = 5 cm, which is less than 6 cm, so a triangle cannot be formed.
9. Check whether side lengths 20 cm, 40 cm and 50 cm can form a triangle.
Yes. All three triangle inequalities hold: 20 + 40 = 60 > 50, 20 + 50 = 70 > 40, and 40 + 50 = 90 > 20. Therefore these lengths can form a triangle.
10. In ΔPQR, ∠P = 80°, ∠Q = 60° and ∠R = 40°. Arrange the sides PQ, QR and RP in order of increasing length.
In a triangle, the larger side is opposite the larger angle. Since ∠R < ∠Q < ∠P, the opposite sides follow the same order: PQ < RP < QR.
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