SECONDARY MATHEMATICS · GEOMETRY PROOF CRAFT
Similarity preserves shape while allowing size to change. That makes it both a proof tool and a scaling engine.
Similar figures share shape, not necessarily size
For similar triangles, corresponding angles are equal and corresponding side lengths are in a common ratio. Unlike congruent triangles, the scale factor need not be 1.
Establish similarity before using ratios
Do not assume two triangles are similar because they look alike. Establish a valid similarity condition from angle equality or proportional side information appropriate to the course.
Once similarity is established, corresponding ratios become justified consequences rather than guesses.
AA is often the cleanest route
If two angles of one triangle equal two corresponding angles of another, the third pair must also be equal because triangle angles sum to 180°. The triangles are similar by AA.
Correspondence controls every ratio
If triangle ABC is similar to triangle PQR with A↔P, B↔Q and C↔R, then AB/PQ = BC/QR = AC/PR. Mixing non-corresponding sides destroys the scale relationship.
Length, area and volume scale differently
If the linear scale factor is k, corresponding lengths scale by k and areas by k². For similar three-dimensional solids, volumes scale by k³.
If a similar triangle is enlarged by linear factor 3/2, an area of 20 cm² becomes 20×(3/2)²=45 cm², not 30 cm².
Similarity can find inaccessible lengths
Parallel lines often create similar triangles inside a larger figure. Once corresponding angles establish similarity, a missing length can be found through proportional sides without direct measurement.
Similarity can prove ratios
A proof may establish two triangles similar first and then use the common scale factor to derive a required ratio. This is stronger than measuring the diagram or assuming proportionality from appearance.
Congruence is a special scale case
Congruent figures can be viewed as similar figures with scale factor 1. Similarity therefore generalises the idea of same shape while relaxing same size.
A similarity proof routine
- Identify the two candidate triangles.
- Establish a valid similarity condition.
- Write corresponding vertices in order.
- State the required proportional sides or equal angles.
- Apply the correct linear, area or volume scale relationship.
Worked scaling checks
A. Linear scale factor 2 → area factor 4.
B. Linear scale factor 3 → volume factor 27 for similar solids.
C. Area ratio 25:9 → positive linear ratio 5:3.
Continue
Build from How to Read a Geometry Diagram, Angle-Chasing Proof and Congruence as a Proof Tool. Return to the Secondary Mathematics Master Index.