Triangles
Introduction
Triangles are the most fundamental geometric figures. Understanding their properties and congruence criteria is essential for geometric proofs in ICSE Class 9.
Congruence of Triangles
Two triangles are congruent if they are exactly the same in shape and size. Corresponding sides and angles are equal.
Notation: △ABC ≅ △DEF means:
- AB = DE, BC = EF, AC = DF
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Congruence Criteria
SSS (Side-Side-Side)
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
<ICSEExample title="SSS Congruence"> If AB = DE = 5 cm, BC = EF = 7 cm, and AC = DF = 8 cm, prove △ABC ≅ △DEF. <Solution> AB = DE (5 cm each) BC = EF (7 cm each) AC = DF (8 cm each) By SSS criterion, △ABC ≅ △DEF </Solution> </ICSEExample>SAS (Side-Angle-Side)
If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
Important: The angle must be between the two sides.
ASA (Angle-Side-Angle)
If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
RHS (Right angle-Hypotenuse-Side)
For right-angled triangles: if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another, the triangles are congruent.
Comparison Table: Congruence Criteria
| Criterion | Conditions | Angle Position |
|---|---|---|
| SSS | All 3 sides equal | Not needed |
| SAS | 2 sides + included angle | Between the sides |
| ASA | 2 angles + included side | Between the angles |
| AAS | 2 angles + non-included side | Anywhere |
| RHS | Hypotenuse + 1 side (right triangles) | Right angle |
Isosceles Triangle Properties
Theorem: Angles opposite equal sides are equal (base angles are equal). Converse: Sides opposite equal angles are equal.
<ICSEExample title="Isosceles Triangle"> In △ABC, AB = AC and ∠B = 50°. Find ∠A and ∠C. <Solution> AB = AC, so base angles are equal. ∠C = ∠B = 50° Sum of angles = 180° ∠A + 50° + 50° = 180° ∠A = 80° </Solution> </ICSEExample>Triangle Inequalities
-
Sum of any two sides > third side
- AB + BC > AC
- BC + AC > AB
- AC + AB > BC
-
Difference of any two sides < third side
-
Larger side has larger angle opposite it
- If AB > AC, then ∠C > ∠B
-
Larger angle has larger side opposite it
- If ∠A > ∠B, then BC > AC
Common Mistakes With Fixes
| Mistake | Correction |
|---|---|
| Using SSA (not a valid criterion) | SSA is NOT a congruence criterion |
| Confusing ASA and AAS | ASA: included side. AAS: non-included side. Both are valid |
| Wrong correspondence when writing congruence | Write vertices in corresponding order |
| Assuming AAA gives congruence | AAA gives similarity, NOT congruence |
ICSE Exam Focus
| Topic | Marks (approx.) | Frequency |
|---|---|---|
| Identifying congruence criteria | 3-4 marks | Very common |
| Proving triangles congruent | 4-5 marks | Very common |
| Isosceles triangle problems | 3-4 marks | Common |
| Triangle inequalities | 2-3 marks | Occasionally asked |
Self-Test
Q1: Which congruence criterion applies if AB = PQ, BC = QR, and ∠B = ∠Q?
Q2: In isosceles △PQR, PQ = PR and ∠Q = 65°. Find ∠P and ∠R.
Q3: Check if sides 5 cm, 7 cm, and 11 cm can form a triangle.
Q4: Prove that the base angles of an isosceles triangle are equal.
Q5: In △ABC, if AB > AC, which angle is greater: ∠C or ∠B?
