Cheat Sheet — Triangles

🔺 Similar Figures & Triangles

ΔABC ~ ΔPQR means
∠A=∠P, ∠B=∠Q, ∠C=∠R AND
AB/PQ = BC/QR = CA/RP

Similar (~)Same shape, sizes may differ. Corresponding angles equal & sides proportional

Congruent (≅)Same shape AND size. A special case of similar with scale factor k = 1

Always similar: any two circles, any two squares, any two equilateral triangles.

Order matters! ΔABC ~ ΔPQR ≠ ΔABC ~ ΔQRP. Vertices must correspond in the written order.

📏 Basic Proportionality (Thales)

BPT: If DE ∥ BC in ΔABC, then
AD/DB = AE/EC

Converse of BPT

If AD/DB = AE/EC, then DE ∥ BC.

Midpoint theorem is a special case: line joining midpoints of two sides is parallel to and half the third side.

Proof idea: triangles between the same parallels have equal areas → the ratio of segments follows.

✅ Criteria for Similarity

Rule	Condition
AAA / AA	All (or two) pairs of corresponding angles equal
SSS	All 3 pairs of corresponding sides in same ratio
SAS	One pair of equal angles & the two including sides proportional

AA is enough: since angles sum to 180°, if two angles match the third automatically does.

No SSA / ASS! Unlike congruence there is no separate RHS/ASA — these three cover all cases.

📐 Areas of Similar Triangles

If ΔABC ~ ΔPQR:
ar(ABC)/ar(PQR) = (AB/PQ)²
= (BC/QR)² = (CA/RP)²

Ratio of areas also equals

(ratio of medians)² = (ratio of altitudes)²
= (ratio of angle bisectors)² = (ratio of perimeters)²

Careful: areas use the SQUARE of the side ratio, but perimeters use the side ratio itself (k, not k²).

📊 Pythagoras Theorem

In right ΔABC with ∠B = 90°:
AC² = AB² + BC²
(hypotenuse)² = sum of squares of legs

Converse

If AC² = AB² + BC², then ∠B = 90°.

Common triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25) — and all their multiples.

⭐ Key Results & Corollaries

Perpendicular from right angle: in a right triangle, the altitude to the hypotenuse creates two triangles each similar to the whole and to each other.

Ratio of perimeters of similar triangles = ratio of corresponding sides = k.

Diagonals of a trapezium ABCD (AB∥DC) divide each other proportionally: AO/OC = BO/OD.

Bisector of an angle of a triangle divides the opposite side in the ratio of the other two sides.

🌍 Common Applications

Heights & ShadowsObject height / shadow length is constant at a given time → find tall object heights

River WidthSimilar triangles find distances that cannot be measured directly

Ladder ProblemsLadder² = wall height² + base distance² (Pythagoras)

Map ScalingActual = map distance × scale factor

Approach: identify the similar triangles or right angle, write the proportion/Pythagoras equation, then solve.

🧠 Exam Quick Recall

Statement	Key Result
BPT	DE∥BC ⇒ AD/DB=AE/EC
Area ratio	= (side ratio)²
Perimeter ratio	= side ratio (k)
Pythagoras	c² = a² + b²
Similarity rules	AA, SSS, SAS

Common mistakes: forgetting to square the ratio for areas; writing wrong vertex correspondence; using SSA (invalid).

Always state the criterion (AA/SSS/SAS) when proving similarity to earn full marks.