To show that two triangles are similar, it is sufficient for one of the following three conditions to hold.
- All three pairs of corresponding sides are proportional (SSS similarity)
- Two pairs of corresponding sides are proportional and the included angles are equal (SAS similarity)
- Two pairs of corresponding angles are equal (AA similarity)
📘 Example ①
In △ABC and △DEF, if ∠A=∠D=50° and ∠B=∠E=70°, which condition shows that △ABC ∽ △DEF?
Answer:Two pairs of corresponding angles are equal (AA similarity). ∠C=∠F=60° are also automatically equal.
In △ABC and △DEF, if ∠A=∠D=50° and ∠B=∠E=70°, which condition shows that △ABC ∽ △DEF?
Answer:Two pairs of corresponding angles are equal (AA similarity). ∠C=∠F=60° are also automatically equal.
💡 Key Point
- AA similarity is used most often (if two angles are equal, all three angles are equal)
- Vertical angles and angles formed by parallel lines can also be used as reasons
Practice Problems
- In △ABC and △DEF, if AB:DE=2:3, BC:EF=2:3, and ∠B=∠E=80°, state the similarity condition.
Answer and Explanation
- Answer:SAS similarity (two pairs of side ratios are equal and the included angles are equal).