Isosceles Triangles
215. Isosceles Triangle Converse · The secret symmetry of isosceles triangles
AB always equals AC when ∠B = ∠C.
Converse of the isosceles triangle theorem: if two angles of a triangle are equal, then the sides opposite them are also equal. So when ∠B = ∠C, it follows that AB = AC — equal angles force equal sides, making the triangle isosceles.
What this lesson covers
Try to break it
Drag A up and down. It glides along the axis of symmetry — the perpendicular bisector of BC — so ∠B always stays equal to ∠C, and AB always comes out equal to AC, no matter how tall or flat you make the triangle. That is the converse of the isosceles theorem: equal base angles force equal sides.
How you build it
Construct an isosceles triangle and see that equal base angles come with equal sides.
- Mark point B — the left end of the base.
- Mark point C — the right end of the base.
- Draw the base segment BC.
- Construct the perpendicular bisector of BC — it crosses BC at its midpoint D.
- Mark the apex A anywhere on the perpendicular bisector — AB and AC come out equal, so triangle ABC is isosceles.
- Draw side AB.
- Draw side AC to complete the isosceles triangle.
The proof, step by step
Prove that if two angles of a triangle are equal, the sides opposite them are equal.
- In ΔABD and ΔACD, ∠B = ∠C (Given)
- ∠ADB = ∠ADC = 90° (By construction, AD ⊥ BC)
- AD = AD (Common side) ⇒ ΔABD ≅ ΔACD (AAS) ⇒ AB = AC (CPCT)
Worked example
In ΔABC, ∠B = ∠C = 50°. If AB = 8 cm, find the length of AC.
Since angles opposite to sides are equal, the sides must be equal. Therefore, AC = AB = 8 cm.
- 4 cm
- 6 cm
- 8 cm — correct
- 10 cm