Concept of Similar Figures
Figures that have the same shape but not necessarily the same size are similar.
Key Takeaways:
Two polygons are similar if:
1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio (proportional).
For similar figures, sizes scale up or down uniformly while all shapes, proportions, and internal angles remain perfectly preserved.
Similarity of Triangles
Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
Proportionality & Similarity Symbol
We write ΔABC ∼ ΔDEF to denote that Triangle ABC is similar to Triangle DEF. Notice that vertex order matters:
• A corresponds to D
• B corresponds to E
• C corresponds to F
The constant ratio of the corresponding sides is called the scale factor or similarity ratio.
AA (Angle-Angle) Similarity Criterion AA
If two angles of one triangle are equal to two angles of another, the triangles are similar.
Why only two angles?
Since the sum of interior angles in any triangle is always 180°, if two angles of one triangle are equal to two angles of another, the third angles must also be equal:
C = 180° - (A + B) and F = 180° - (D + E).
Therefore, the AA criterion is mathematically identical to the AAA criterion.
SSS (Side-Side-Side) Similarity Criterion SSS
If corresponding sides of two triangles are proportional, their corresponding angles are equal, and the triangles are similar.
Law of Cosines Verification
Even though we only set side lengths, the angles are locked automatically by the Law of Cosines:
cos(A) = (b² + c² - a²) / 2bc.
Since the sides are scaled proportionally (a' = k·a, b' = k·b, c' = k·c), the scale factor k² cancels out in the numerator and denominator, leaving the angles completely unchanged.
SAS (Side-Angle-Side) Similarity Criterion SAS
If one angle of a triangle equals one angle of another triangle, and the sides containing these angles are proportional, the triangles are similar.
Included Angle Requirement
It is vital that the angle is included between the proportional sides. If the angle is not between the proportional sides (SSA), similarity is not guaranteed, and multiple distinct triangles could be formed.