Evidence, similarity, consequence
Most similarity problems have three stages: establish usable geometric facts, prove triangles similar with a named criterion, then use corresponding parts to obtain the requested length or relation.
Learning goals
- Extract usable angle, parallel-line and ratio evidence from a diagram.
- Build a justified similarity chain.
- Use similarity to find lengths, ratios, perimeters and indirect heights.
- Prove a derived geometric relation using similar triangles.
Concept lesson
Read an ordered similarity statement
If triangle ABC is similar to triangle DEF, then A↔D, B↔E, C↔F and AB/DE=BC/EF=CA/FD. Keep this order fixed.
Criterion evidence checklist
Look for common, vertically opposite, alternate or corresponding angles; parallel lines; and proportional side data. Use only facts given or justified.
Annotate before solving
Mark every given parallel pair, equal angle, shared angle, right angle and known side ratio. Then identify candidate triangle pairs; do not infer equality or parallelism from how the drawing looks.
Turn similarity into the target relation
After proving similarity, write only the corresponding-parts equality that can lead to the target. Substitute known equalities or segment sums, cross multiply if required, and explicitly state the requested conclusion.
Scale factor and perimeter
Corresponding lengths and perimeters share the same linear scale factor. Establish its direction from one known corresponding pair before applying it to another side or perimeter.
Extract hidden equal angles
In intersecting or nested-triangle diagrams, identify vertically opposite angles, shared angles and angles created by parallel lines. Attach a named reason to each equality before choosing AA similarity.
Build a complete similarity chain
Write two justified angle equalities, state the ordered AA similarity conclusion, select the corresponding-side proportion containing the target, substitute any given equal segments, and algebraically derive the required relation.
Missing length and perimeter from one scale factor
If corresponding sides 6 cm and 9 cm establish a scale factor of 3/2 from the first triangle to the second, multiply every first-triangle length and its perimeter by 3/2. Do not reverse the factor midway.
Finish the proof after similarity
Given a partially completed proof ending with similar triangles, choose the relevant corresponding-parts equality and complete the algebra to reach the stated geometric relation.
Audit every ratio
Inspect four proposed proportions, reject those with a mismatched pair or reversed direction, then calculate the requested side using a corrected proportion.
From diagram to final answer
Annotate an unfamiliar diagram, prove the intended triangle pair similar, map corresponding sides and calculate a missing length. Each stage is scored independently for analytics.
Find height using simultaneous shadows
A vertical object and a vertical reference stick cast shadows at the same time. Use equal sun angles and right angles to prove the height-shadow triangles similar, then solve height/reference-height = shadow/reference-shadow.
Two- and three-mark proof discipline
Earn separate evidence credit by writing angle or ratio facts with reasons, naming the criterion, ordering the similarity statement correctly, and using corresponding parts to reach the requested result.
Integrated similarity mastery
Solve one unfamiliar diagram problem and one shadow problem. For each, identify evidence, justify similarity, preserve correspondence and complete the requested numerical or algebraic conclusion.