Use only the evidence you actually have
A similarity criterion is a sufficient evidence pattern. Identify whether the diagram supplies angles, three side ratios, or two side ratios with their included angle before naming AAA/AA, SSS or SAS.
Learning goals
- Select AAA/AA, SSS or SAS from the available evidence.
- Verify every condition required by the selected criterion.
- Write a correspondence-consistent similarity conclusion.
- Reproduce the prescribed AAA, SSS and SAS theorem proofs.
Concept lesson
Correspondence before criterion
Place corresponding vertices in columns. Every angle pair and side pair in the proof must follow that single map.
Check proportional sides
Reduce or cross-multiply corresponding side ratios. All ratios required by the named criterion must be equal.
AAA/AA, SSS and SAS decision table
AAA/AA: two corresponding angles equal, so the third is equal. SSS: all three corresponding side pairs proportional. SAS: two corresponding side pairs proportional and the included angles equal. Do not use a criterion with missing evidence.
The included angle test for SAS
Circle the two proportional sides in each triangle. Their common vertex identifies the included angle. Only equality of those two included angles completes SAS similarity.
Architecture of a theorem proof
Write given, to prove and construction; invoke the earlier theorem used in the construction; establish the required equality or proportionality; then conclude similarity with the named criterion and correct order.
Verify SSS without skipping the third pair
For sides 4, 6, 8 and 6, 9, 12, verify 4/6=6/9=8/12=2/3. Match shortest-to-shortest, middle-to-middle and longest-to-longest, then write the similarity statement in that order.
Prove similarity by AA
Establish one angle pair using vertically opposite angles and another using alternate interior angles from parallel lines. Map the equal-angle vertices, conclude AA similarity, and preserve that order in every subsequent ratio.
Three prescribed theorem-proof routes
For AAA, use two equal-angle pairs and angle sum for the third. For SSS, construct a smaller triangle parallel to a side, obtain proportional divisions through BPT, then prove congruence to transfer the remaining sides. For SAS, use the proportional-side construction, BPT and congruence while keeping the given included angle central. Practise each proof with every reason stated.
Evidence-to-conclusion chain
Complete missing reasons in three short proofs and then write the similarity conclusion. A submission is accepted only when the conclusion order matches every evidence pair.
Reject insufficient criteria
Classify evidence cards as sufficient or insufficient. Reject two side ratios plus a non-included angle as SAS, and reject only two side ratios as SSS; name the exact missing fact.
Rotate the diagram, preserve the proof
Prove the same pair of triangles similar in two orientations. Use labels and evidence, not visual position, to preserve the correspondence and criterion.
Choose the shortest valid proof
When a diagram supports more than one route, select the criterion requiring the fewest derived facts, verify its exact conditions, and write the correspondence-consistent conclusion.
One-mark recognition and four-mark proof
For one mark, name the criterion only after checking its defining evidence. For four marks, reproduce the prescribed theorem argument with construction, intermediate theorem use and reasons; the criterion name alone is not a proof.
Criteria mastery check
Identify criteria in three diagrams, reject one insufficient evidence set, write every conclusion in correct order, and complete one randomly selected prescribed theorem proof with reasons.