Correspondence controls every conclusion
A triangle may be turned, reflected or drawn at another scale. Match vertices by equal angles and connected corresponding sides, not by where they appear on the page.
Learning goals
- Match corresponding vertices, angles and sides in similar triangles.
- Write a similarity statement in consistent corresponding order.
- Use equal corresponding angles and proportional corresponding sides to find unknown measures.
Concept lesson
Meaning of similar triangles
In similar triangles, corresponding angles are equal and corresponding sides are proportional.
Equivalent ratios
Equivalent ratios preserve the order of compared quantities. Solve a/b = c/d by cross multiplication only after checking that a corresponds to c and b to d.
Build a correspondence map
If angle A equals angle D, angle B equals angle E and angle C equals angle F, write A↔D, B↔E, C↔F. This gives AB↔DE, BC↔EF and CA↔FD, so the consistent statement is triangle ABC similar to triangle DEF.
One scale factor links all corresponding sides
For triangle ABC similar to triangle DEF, AB/DE = BC/EF = CA/FD. Each ratio uses a side from the first triangle over its partner from the second triangle; reversing every ratio is also valid, but mixing directions is not.
Write the correct similarity statement
Given angle P = angle Y, angle Q = angle Z and angle R = angle X, first write P↔Y, Q↔Z, R↔X. Therefore triangle PQR is similar to triangle YZX; PQ↔YZ, QR↔ZX and RP↔XY.
Find a corresponding side
If triangle ABC is similar to triangle DEF, AB=6 cm, DE=9 cm and BC=8 cm, then AB/DE = BC/EF gives 6/9 = 8/EF, so EF=12 cm. The scale factor from ABC to DEF is 3/2.
Match after rotation
Use equal-angle marks on two differently oriented triangles to complete a vertex map, a side map and one valid similarity statement. Verify each adjacent side pair before submitting.
Keep ratios in one direction
From a supplied similarity statement, mark each corresponding pair, reject two inconsistent proportions, then use a consistent proportion to calculate an unknown side.
Scale drawing of a triangular plot
A triangular plot and its scale drawing have matching vertices. Use the known drawing-to-ground side pair to determine the scale factor and calculate another ground length.
One-mark correspondence check
Before calculating, copy the similarity statement and place corresponding vertices in columns. Most one-mark errors here come from using the right calculation on the wrong side pair.
Correspondence and scale exit ticket
For two rotated similar triangles, construct the correspondence map, write a consistent similarity statement, calculate one missing side and explain why a visually adjacent but non-corresponding ratio is invalid.