Study Notes: Triangles (SOF_IMO)

Overview

Triangles form the foundation of geometry and appear in roughly 10–15% of SOF IMO questions. You'll face problems on congruence (proving triangles are identical), properties of special triangles (especially isosceles), and triangle inequalities (which side lengths are possible). This topic connects directly to quadrilaterals, constructions, and coordinate geometry, so mastering it unlocks several downstream chapters.

The exam tests both computational skills (finding angles, sides) and logical reasoning (proving congruence or disproving possible triangles). You must memorize four congruence criteria, know the angle-side relationships in isosceles and equilateral triangles, and apply the triangle inequality theorem fluently. Questions often combine multiple concepts—for example, proving congruence to find an unknown angle, then using that angle in an inequality check.

Strong performance here requires recognizing patterns in diagrams, choosing the correct congruence criterion quickly, and avoiding common sign errors in inequality problems. Practice marking equal sides/angles on figures and writing two-column proofs to build speed.

Key Concepts

• **Congruent triangles** are identical in shape and size. All corresponding sides and angles are equal. Symbol: △ABC ≅ △PQR means AB = PQ, BC = QR, CA = RP and all corresponding angles match.
• **Four congruence criteria**: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and RHS (right angle, hypotenuse, one side). No AAA criterion—similar triangles can have different sizes.
• **CPCT principle**: Corresponding Parts of Congruent Triangles are equal. After proving congruence, you can state any corresponding element is equal without further proof.
• **Isosceles triangle**: Two sides equal implies two base angles equal. Conversely, two angles equal implies opposite sides equal. The altitude from the apex bisects the base and is perpendicular to it.
• **Equilateral triangle**: All sides equal, all angles 60°. Every altitude is also a median, angle bisector, and perpendicular bisector.
• **Triangle inequality theorem**: Sum of any two sides must exceed the third side. For sides a, b, c: a + b > c, b + c > a, c + a > b. Also, difference of two sides is less than the third: |a − b| < c.
• **Angle-side relationship**: In any triangle, the side opposite the largest angle is longest; the side opposite the smallest angle is shortest. This extends to inequalities: if angle A > angle B, then side opposite A (BC) > side opposite B (AC).
• **Exterior angle theorem**: An exterior angle equals the sum of the two non-adjacent interior angles. The exterior angle is greater than either remote interior angle.

Formulas / Key Facts

1. **Sum of angles in a triangle** = 180°. If two angles are known, the third is 180° minus their sum. 2. **SSS congruence**: If AB = PQ, BC = QR, CA = RP, then △ABC ≅ △PQR. 3. **SAS congruence**: If AB = PQ, angle B = angle Q, BC = QR, then △ABC ≅ △PQR (angle must be included between the two sides). 4. **ASA congruence**: If angle A = angle P, AB = PQ, angle B = angle Q, then △ABC ≅ △PQR. 5. **RHS congruence**: For right triangles, if hypotenuse and one side are equal, triangles are congruent. 6. **Isosceles triangle property**: If AB = AC, then angle B = angle C (and vice versa). 7. **Triangle inequality**: For sides a, b, c: a + b > c, b + c > a, c + a > b. Equivalently, any side < sum of other two sides. 8. **Difference inequality**: The third side must be greater than the difference of the other two: c > |a − b|. 9. **Pythagorean theorem (right triangles only)**: a² + b² = c², where c is the hypotenuse.

Worked Examples

**Example 1: Congruence proof** Given: In △ABC and △DEF, AB = DE = 5 cm, BC = EF = 7 cm, and angle B = angle E = 50°. Prove △ABC ≅ △DEF and find angle A if angle D = 60°.

*Solution:*

• AB = DE, BC = EF (given), angle B = angle E = 50° (given).
• Angle B is included between sides AB and BC; angle E is included between DE and EF.
• By SAS criterion, △ABC ≅ △DEF.
• By CPCT, angle A = angle D = 60°.

**Example 2: Isosceles triangle** In △PQR, PQ = PR = 8 cm and angle Q = 55°. Find angle R and angle P.

*Solution:*

• PQ = PR means △PQR is isosceles with base QR.
• Base angles are equal: angle Q = angle R = 55°.
• Sum of angles = 180°, so angle P = 180° − 55° − 55° = 70°.

**Example 3: Triangle inequality** Can a triangle have sides 4 cm, 7 cm, and 12 cm?

*Solution:*

• Check: 4 + 7 = 11, which is NOT greater than 12.
• Since 4 + 7 ≤ 12, the triangle inequality fails.
• Answer: No, such a triangle cannot exist.

**Example 4: Finding possible side length** Two sides of a triangle are 5 cm and 9 cm. What is the range for the third side x?

*Solution:*

• By triangle inequality: 5 + 9 > x → x < 14.
• Also: x + 5 > 9 → x > 4.
• Range: 4 cm < x < 14 cm (x must be strictly between 4 and 14).

Common Mistakes

1. **Confusing SAS with SSA** → SSA (two sides and a non-included angle) is NOT a congruence criterion. The angle must be between the two sides. *Fix:* Check that the angle is sandwiched between the given sides.

2. **Using AAA to prove congruence** → Three equal angles mean similar triangles, not necessarily congruent (they could be different sizes). *Fix:* You need at least one side length to prove congruence.

3. **Forgetting the strict inequality** → Writing a + b ≥ c instead of a + b > c. A degenerate triangle (collinear points) where a + b = c is not a valid triangle. *Fix:* Always use > not ≥.

4. **Mixing up base and equal sides in isosceles triangles** → Assuming the base is one of the equal sides. *Fix:* The base is the third side (the one that's different); the two equal sides meet at the apex.

5. **Applying RHS to non-right triangles** → RHS only works for right-angled triangles. *Fix:* Verify one angle is 90° before using RHS criterion.

Quick Reference

• **Congruence shorthand**: SSS, SAS, ASA, RHS (no AAA or SSA).
• **Isosceles rule**: Equal sides → equal opposite angles; altitude from apex bisects base at 90°.
• **Equilateral triangle**: All sides equal, all angles 60°.
• **Triangle inequality**: Sum of two sides > third side; difference of two sides < third side.
• **Exterior angle = sum of remote interior angles** (and > either one alone).
• **CPCT**: After proving congruence, all corresponding parts are automatically equal.