Math101 · Independent Ontario learning library
Math101Triangle Similarity
Similar triangles have equal corresponding angles and proportional corresponding side lengths. They share shape but may differ in scale.
Similarity supports indirect measurement, maps, models, trigonometry, and proofs. Scale-factor powers connect one-, two-, and three-dimensional measures.
Intuition and core definition
Similar triangles have equal corresponding angles and proportional corresponding side lengths. They share shape but may differ in scale. Sufficient criteria are AA, SSS when all three corresponding side ratios are equal, and SAS when two corresponding side ratios are equal and their included angles are congruent.
Notation, language, and conditions
$\triangle ABC\sim\triangle DEF$ fixes vertex correspondence. A scale factor from first to second is $k=DE/AB=EF/BC=DF/AC$. Ratios must compare corresponding sides in a consistent direction; perimeters scale by $k$ and areas by $k^2$.
Why this idea matters
Similarity preserves angles and fixes one scale factor across every pair of corresponding lengths, with area changing by the factor's square.
A dependable method
- Establish similarity using AA, SSS proportionality, or SAS proportionality.
- Write the similarity statement in matched vertex order.
- Pair corresponding sides in a table.
- Set a proportion using one consistent image/preimage direction and solve.
- Check all available ratios and angle correspondence.
