Triangles

Why This Chapter Matters

Triangles are the building blocks of plane geometry. Complex figures can often be split into triangles, and once two triangles are shown congruent, many hidden equalities become available.

The central skill is proof by matching. Identify two triangles, compare corresponding parts, choose a valid congruence criterion and only then use CPCT to deduce further equal sides or angles.

The second half of the chapter develops inequalities. Here the focus shifts from exact equality to comparison: longer sides face larger angles, and side lengths must satisfy the triangle inequality. Together, congruence and inequality form a powerful geometric toolkit.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Angle sum of a triangle
• Basic line-angle results
• Corresponding parts
• Measurement of sides and angles

These are the tools this chapter reuses and deepens.

1. Congruence vs Similar Appearance

Two triangles are congruent when one can be placed exactly over the other. Their corresponding sides and angles are equal.

A drawing can look equal without being mathematically congruent. Proof requires enough information to guarantee a unique triangle. This is why formal congruence criteria matter.

2. SSS and SAS

SSS states that if the three sides of one triangle equal the three corresponding sides of another, the triangles are congruent.

SAS requires two sides and the included angle. The word included is essential: the angle must lie between the two known sides. Two sides and a non-included angle do not generally determine one unique triangle.

3. ASA, AAS and RHS

ASA uses two angles and the included side. Because the third angle is fixed once two angles are known, an AAS arrangement also establishes congruence.

RHS applies specifically to right triangles: equal hypotenuse and one corresponding side are enough. The right angle supplies the missing fixed-angle information.

4. CPCT

CPCT means corresponding parts of congruent triangles are equal.

It is not a congruence criterion. First prove the triangles congruent; then use CPCT for additional side or angle equalities.

Write the congruence statement in matching order. If ΔABC ≅ ΔPQR, then A↔P, B↔Q and C↔R.

5. Isosceles Triangle Theorems

If two sides of a triangle are equal, the angles opposite those sides are equal. The converse is also true: equal angles stand opposite equal sides.

These theorems convert side information into angle information and vice versa. They are especially useful when congruence has already produced one equality.

6. Triangle Inequality

For any triangle, the sum of any two sides is greater than the third side.

This is a feasibility test. Lengths 2, 3 and 6 cannot form a triangle because 2+3 is not greater than 6.

The condition reflects geometry: the direct side between two points must be shorter than a broken path through the third vertex.

7. Side-Angle Relationship

The larger side lies opposite the larger angle, and the larger angle lies opposite the larger side.

This allows comparison without measurement. If ∠A>∠B, then side BC>AC because BC is opposite ∠A and AC is opposite ∠B.

8. Proof Strategy

Use a fixed order: identify the two triangles, list equal parts with reasons, choose a valid congruence rule, write the congruence statement in correct order, and then apply CPCT if needed.

This disciplined approach makes even crowded diagrams manageable.

Visualise the Core Idea

Congruence works only when the given measurements determine one unique triangle.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary