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Math101learn.math101.caTriangle 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.
Worked example
Representations and interpretation
A dilation followed by rigid motions maps one similar triangle to the other. Correspondence tables show equal angles and constant length ratios; grids reveal the scale factor visually.
Reasoning about variations
Equal side ratios alone require all three corresponding pairs, while two side ratios require the included angle for SAS. SSA-like similarity data can also be ambiguous.
Common mistakes
How to check your work
- Verify at least two independent side ratios.
- Confirm the similarity statement matches every angle pair.
- Estimate whether the solved length fits the enlargement or reduction.
Practice
- A side $4$ corresponds to $10$. What is the enlargement scale factor?
- Two triangles share two equal angle measures. Which criterion proves similarity?
- If length scale factor is $3$, what is area factor?
Answers and brief solutions
Show answers
- $2.5$ $10/4=2.5$.
- AA The third angles then also match.
- $9$ Area scales by $3^2$.
Synthesis and transfer
Indirectly measuring a tree from its shadow uses AA similarity; matching sun angles establishes shape before a consistent corresponding-side ratio yields the height.
A $1.5$ m post casts a $2$ m shadow while a tree casts a $12$ m shadow under the same sunlight. Equal elevation angles and right angles establish AA similarity, so tree height $h$ satisfies $h/12=1.5/2$, giving $9$ m. The shadow correspondence must remain consistent; reversing only one ratio would break the proportion. If the sun angle changes between measurements, AA no longer follows and the model fails. The area ratio of the triangles would be the square of the length factor, showing why similarity preserves shape but not size-dependent quantities such as area.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A side $4$ corresponds to $10$. What is the enlargement scale factor?
- $10/4=2.5$.
End of lesson
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