“If the base angles are equal, the triangle is isosceles” (the converse also holds). A triangle whose three sides are all equal is called an equilateral triangle, and each angle is 60°.
Example ①
In △ABC, when ∠B=∠C, prove that AB=AC.
Answer: Let M be the intersection of the bisector of ∠A and BC. In △ABM and △ACM / ∠B=∠C (given) / ∠BAM=∠CAM (bisector) / AM=AM (common side) / By ASA, △ABM≅△ACM. Therefore, AB=AC.
In △ABC, when ∠B=∠C, prove that AB=AC.
Answer: Let M be the intersection of the bisector of ∠A and BC. In △ABM and △ACM / ∠B=∠C (given) / ∠BAM=∠CAM (bisector) / AM=AM (common side) / By ASA, △ABM≅△ACM. Therefore, AB=AC.
Key Points
- “Equal base angles → isosceles triangle” (converse theorem)
- Equilateral triangle: three equal sides → all three angles are 60°
- All three angles are 60° → three equal sides
Practice Problem
- In equilateral triangle ABC, point D on side BC and point E on side CA satisfy BD=CE. Prove that △ABD ≅ △BCE and show that AD=BE.
Answers and Explanations
- Answer: In △ABD and △BCE / AB=BC (equilateral triangle) ① / BD=CE (given) ② / ∠ABD=∠BCE=60° (equilateral triangle) ③ / By SAS, △ABD≅△BCE. Therefore, AD=BE.