Write a proof of similarity in the same format as a proof of congruence. Use a similarity condition as the basis.
📘 Example ①
In △ABC, let H be the foot of the perpendicular drawn from vertex A to BC. Prove that △ABH ∽ △ACB.
Answer:For △ABH and △ACB / ∠AHB = ∠ABC = 90° (given condition and right angles) ① / ∠B = ∠B (common angle) ② / From ① and ②, two pairs of corresponding angles are equal, so △ABH ∽ △ACB (AA similarity).
In △ABC, let H be the foot of the perpendicular drawn from vertex A to BC. Prove that △ABH ∽ △ACB.
Answer:For △ABH and △ACB / ∠AHB = ∠ABC = 90° (given condition and right angles) ① / ∠B = ∠B (common angle) ② / From ① and ②, two pairs of corresponding angles are equal, so △ABH ∽ △ACB (AA similarity).
💡 Key Point
- The proof format is the same as for congruence: begin with “For △○○ and △□□”
- When using AA similarity: write the reasons for the two pairs of equal angles
- Finally, write “Since two pairs of corresponding angles are equal, △○○ ∽ △□□”
Practice Problems
- In △ABC, when ∠BAC=90° and AD⊥BC (D lies on BC), prove that △ABD ∽ △CAD.
Answer and Explanation
- Answer:For △ABD and △CAD / ∠ADB = ∠ADC = 90° (AD⊥BC) / ∠ABD = ∠CAD (because ∠BAC=90°, ∠ABD+∠BAD=90°=∠CAD+∠BAD) / Therefore, by AA, △ABD ∽ △CAD.