Similarity criteria for triangles

1. Definition of similarity: proportional sides and equal angles

Definition of Similarity

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional. This means that one triangle is a scaled version of the other, preserving shape but not necessarily size.

Intuition: Similar triangles have the same shape but different sizes, like a photograph and its enlargement.

Core Rules:

• All three pairs of corresponding angles must be equal: $\angle A = \angle A'$, $\angle B = \angle B'$, $\angle C = \angle C'$
• All three pairs of corresponding sides must have the same ratio: $\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'} = k$, where $k$ is the scale factor
• The scale factor $k$ is constant for all corresponding side pairs
• Both conditions must hold simultaneously for similarity

Consequence: If triangles are similar, knowing one side length in each triangle determines the scale factor for all other measurements.

Example: If triangle ABC has sides 3, 4, 5 and triangle DEF has sides 6, 8, 10, then $k = 2$ and the triangles are similar.

2. Angle-Angle (AA) similarity criterion

Angle-Angle (AA) Similarity Criterion

Two triangles are similar if two pairs of corresponding angles are equal. The third angle pair is automatically equal because the angle sum in any triangle is $180^\circ$.

Intuition: Knowing two angles fixes the triangle's shape completely, so matching two angles guarantees the triangles have the same shape.

Core Rules:

• Only two angle equalities are needed to prove similarity
• The third angle equality follows from the angle sum property: if $\angle A = \angle A'$ and $\angle B = \angle B'$, then $\angle C = 180^\circ - \angle A - \angle B = \angle C'$
• AA is the most commonly used similarity criterion in practice
• Once AA similarity is established, corresponding sides are automatically proportional

Consequence: AA provides the quickest path to proving similarity, requiring only angle measurements without needing side lengths.

Example: If triangle ABC has angles $50^\circ$, $60^\circ$, $70^\circ$ and triangle DEF has angles $50^\circ$, $60^\circ$, then the triangles are similar by AA.

3. Side-Side-Side (SSS~) and Side-Angle-Side (SAS~) similarity theorems

Side-Side-Side and Side-Angle-Side Similarity Theorems

SSS~ Theorem: Two triangles are similar if all three pairs of corresponding sides are proportional. SAS~ Theorem: Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are equal.

Intuition: SSS~ checks if one triangle is a perfect scaled copy by comparing all side ratios. SAS~ combines one angle match with two side ratios around that angle.

Core Rules (SSS~):

• Verify $\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'}$
• All three ratios must be exactly equal

Core Rules (SAS~):

• Two side ratios must be equal: $\frac{AB}{A'B'} = \frac{AC}{A'C'}$
• The angle between these sides must be equal: $\angle A = \angle A'$

Consequence: These criteria allow proving similarity using only side measurements (SSS~) or a mix of one angle and two sides (SAS~).

Example: Triangles with sides 4, 6, 8 and 6, 9, 12 are similar by SSS~ since $\frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{2}{3}$.

4. Solving for missing sides in similar triangles using proportions

Solving for Missing Sides Using Proportions

Once similarity is established, unknown side lengths are found by setting up and solving proportions using the constant scale factor between corresponding sides.

Intuition: The scale factor acts as a conversion ratio between the two triangles, allowing calculation of any unknown side from its corresponding known side.

Core Rules:

• Identify corresponding sides carefully (matching vertices matter)
• Set up proportion: $\frac{\text{side in triangle 1}}{\text{corresponding side in triangle 2}} = \frac{\text{another side in triangle 1}}{\text{corresponding side in triangle 2}}$
• Cross-multiply to solve: if $\frac{a}{b} = \frac{c}{x}$, then $ax = bc$, so $x = \frac{bc}{a}$
• Verify the scale factor is consistent across all known sides

Consequence: Similar triangles transform geometric problems into algebraic proportion-solving, enabling indirect measurement of inaccessible distances.

Example: If triangles are similar with sides 5, 7, $x$ and 10, 14, 16, then $\frac{5}{10} = \frac{x}{16}$ gives $x = 8$.

5. Applications: Indirect measurement and scale modeling

Applications: Indirect Measurement and Scale Modeling

Shadow Method: An object's height is determined by comparing its shadow length to a reference object's shadow and height, using similar triangles formed by the sun's rays. Scale Modeling: Architectural models use similarity to represent buildings at reduced scale while preserving proportions.

Intuition: Similar triangles allow measuring unreachable heights or distances by working with accessible, proportional measurements.

Core Rules (Shadow Method):

• Measure shadow lengths at the same time (ensures equal sun angles)
• Set up proportion: $\frac{\text{object height}}{\text{object shadow}} = \frac{\text{reference height}}{\text{reference shadow}}$

Core Rules (Scale Modeling):

• Choose scale factor (e.g., 1:100 means model is $\frac{1}{100}$ actual size)
• All dimensions scale by the same factor
• Area scales by $k^2$ and volume by $k^3$

Consequence: Similarity transforms impractical direct measurements into solvable proportion problems.

Example: A 2-meter stick casts a 3-meter shadow while a tree casts a 12-meter shadow. Tree height: $\frac{h}{12} = \frac{2}{3}$ gives $h = 8$ meters.