L5 · Triangle problems clinic

Learning goal

“I can look at any fully described pair of triangles in this unit and choose the right verdict and tool without prompting - congruent (SSS/SAS/ASA), similar (AA/SSS/SAS), or neither - and apply the proportionality and altitude theorems in real contexts like shadow measurement, stating correspondence correctly every time.”

Included — "between": the angle between two sides, the side between two angles; the word every criterion hinges on (from L1).

Corresponding — matched by the statement's positions, the only safe way to pair parts (from L2).

Proportional — ratios unanimous across all pairs, or the claim fails (from L3).

Similar but not congruent — same shape, scale factor not equal to 1: the verdict students most often under-report (from L3).

Worked example

Worked example 1. Triangle ABC has angle A = 60 degrees, AB = 10, angle B = 45 degrees. Triangle DEF has angle D = 60 degrees, DE = 10, angle E = 45 degrees. Two angle pairs equal and the included side equal: congruent by ASA. (Also similar, trivially - congruence is the stronger, reportable verdict.) Worked example 2. One pair of triangles has sides 7, 10, 13; the other has sides 14, 20, 26. Ratios all 1/2: similar by SSS, scale factor 2 - but NOT congruent: the sizes differ. Worked example 3. A third pair shares one pair of equal angles and one pair of equal sides, nothing more. Verdict: NEITHER can be concluded - two facts, no criterion. Say it plainly: sort every problem by its data before computing. Angles only - similarity possible (AA), congruence impossible. Sides only - ratio audit decides similarity; equality audit decides congruence. Mixed data - hunt for a criterion's exact inventory, included-ness included.

Check yourself — explain it

A pair of triangles is found congruent. A classmate asks for their similarity verdict too. The complete answer is...

Paired problem — example still visible

Triangle ABC has sides 9, 12, 15. Triangle DEF has sides 9, 12, 15. Verdict? -> Congruent by SSS (and hence similar, scale factor 1)

Faded completion — last step is yours

"All angles equal in pairs, sizes different: the verdict is ___ but not ___" -> similar; congruent

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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