Triangles and Congruence: Class 9 Study Guide
Two triangles are congruent when they have exactly the same shape and size.
Two triangles are congruent if one can be placed exactly on the other. Their matching sides and angles are equal. We write the triangles with the matching vertices in the same order: △ABC ≅ △PQR means A matches P, B matches Q and C matches R.
We do not need to check all six parts. Certain sets of three are enough, called congruence rules.
Congruence rules
| Rule | What must match |
|---|---|
| SSS | Three sides |
| SAS | Two sides and the angle between them |
| ASA | Two angles and the side between them |
| AAS | Two angles and a side not between them |
| RHS | Right angle, hypotenuse and one side (right-angled triangles) |
Isosceles triangles and inequality
In an isosceles triangle, the angles opposite the equal sides are equal, and the converse is also true. The sum of any two sides of a triangle is greater than the third side. So sides 3, 4 and 8 cannot form a triangle, because 3 + 4 = 7 is less than 8. Sides 3, 4 and 5 can.
Use tick marks.
SSS, SAS, ASA, AAS or RHS.
Keep the vertex order.
Corresponding parts are equal.
Using congruence
To prove that the base angles of an isosceles triangle are equal, draw the bisector of the apex angle. The two small triangles are congruent by SAS, because they share the bisector, have two equal sides and equal angles between them. So the base angles are equal, by corresponding parts.
Worked examples and common mistakes
Example: In △ABC, AB = AC and ∠B = 40°. Then ∠C = 40° and ∠A = 100°. Example: Can 5 cm, 6 cm and 12 cm form a triangle? No, since 5 + 6 = 11 is less than 12. Common mistakes: using 'SSA' (not a valid rule), writing the vertices in the wrong order, and assuming a figure is to scale. Use only given facts.
Check your understanding: model answers
1. SAS. 2. Yes, because 7 + 8 = 15 is more than 10 and the other sums are larger. 3. (180° − 50°) ÷ 2 = 65°.
✨ Key Takeaways
- Congruent triangles have equal matching sides and angles.
- Use SSS, SAS, ASA, AAS or RHS.
- Angles opposite equal sides are equal.
- Any two sides together exceed the third.
💬 Questions for Discussion & Reflection
- Which rule proves two triangles congruent if two sides and the included angle match?
- Can 7, 8 and 10 form a triangle?
- In an isosceles triangle the apex angle is 50°. Find a base angle.