Part A · FoundationClass 8 revision

Congruent Triangles തുല്യത്രികോണങ്ങൾ

Two triangles are congruent ("equal triangles") when they are exact copies of each other – the same shape and the same size. This page revises the Class 8 chapter, and ends with the question that opens Chapter 6: what if two triangles have the same shape but different sizes?

1. What does "equal triangles" mean?

Cut two triangles from paper. If you can place one exactly on top of the other – after sliding it, turning it, and if necessary flipping it over – they are congruent (തുല്യത്രികോണങ്ങൾ). We write ΔABC ≅ ΔPQR. Every side and every angle of one equals the matching side and angle of the other.

🎬 Slide, turn, flip – do they fit?

2. Matching parts (corresponding parts)

When we write ΔABC ≅ ΔPQR, the order of the letters tells us which vertex matches which: A ↔ P, B ↔ Q, C ↔ R. So AB = PQ, BC = QR, CA = RP and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

👀 Equal marks show the matching parts

One tick, two ticks, three ticks mark equal sides; the coloured arcs mark equal angles. Six matching parts in total – three sides and three angles.

3. Do we need all six? The four congruence tests

Good news: you never need to check all six parts. Three well-chosen measurements are enough to fix a triangle completely. In each lab below, move the sliders – whatever values you choose, the data builds exactly one triangle, so any two people using the same data get congruent triangles.

🎚️ SSS – three Sides

BC = a CA = b AB = c

Three sides fix all three angles – a triangle with given sides is rigid. (Sides must obey the triangle inequality.)

🎚️ SAS – two Sides and the Angle between them

AB = c AC = b ∠A

The angle must be the one between the two given sides (the included angle).

🎚️ ASA – two Angles and the Side between them

AB = c ∠A ∠B

Since the third angle is 180° minus the other two, "two angles and any one side" (AAS) also works.

🎚️ RHS – Right angle, Hypotenuse and one Side

hypotenuse AB leg BC

Pythagoras fixes the third side, so the triangle is fixed.

⭐ The four tests for congruent triangles

SSS

three sides equal

SAS

two sides and the included angle equal

ASA / AAS

two angles and one side equal

RHS

right angle, hypotenuse and one side equal

4. Two tests that do NOT work

SSA – two sides and a non-included angle

If the angle is not between the two sides, the data can build two different triangles. Watch the side BC swing like a compass: it can meet the other side in two places.

🎚️ SSA is ambiguous

AAA – three angles

Three equal angles give triangles with the same shape, but they can be any size. These triangles are not congruent. Yet they clearly have something in common…

🎚️ Same angles, many sizes

Size of the last triangle

Every triangle here has angles 40°, 60°, 80°. Same shape, any size – this is exactly what Chapter 6 studies. Such triangles are called similar (സദൃശം), and the chapter finds the precise rule connecting their sides.

💡 The bridge to Chapter 6

Congruent = same shape and same size (sides equal). Similar = same shape, size may differ (sides in the same ratio). Congruent triangles are just similar triangles with scale factor 1.

5. Isosceles triangles: equal sides ⇔ equal angles

An isosceles triangle (സമപാർശ്വത്രികോണം) has two equal sides. The angles opposite those sides – the base angles – are equal. And the reverse is true: if two angles are equal, the sides opposite them are equal. (Used in Exercise 1, Question 7.)

🖐️ Drag C – when are ∠A and ∠B equal?

🎬 Why? Fold it along the bisector

6. Check yourself

🧭 Summary

Congruent triangles: same shape and same size; one fits exactly on the other.
Tests: SSS, SAS (included angle), ASA/AAS, RHS. Three well-chosen parts fix the triangle.
SSA can give two triangles; AAA gives the same shape at any size – not congruent, but similar.
Isosceles: two equal sides ⇔ two equal base angles.

Next: Parallel lines and how they divide the sides of a triangle ▶