Isosceles Triangles
214. Isosceles Triangle Theorem · equal sides always hide equal angles
The base angles ∠B and ∠C remain exactly equal as you reshape the triangle.
Isosceles triangle theorem: the angles opposite the equal sides (the base angles) are equal. So in triangle ABC with AB = AC, the base angles ∠B = ∠C, however the triangle is reshaped while the two sides stay equal.
What this lesson covers
Try to break it
Drag B to widen or narrow the base BC. The altitude AD adjusts to stay perpendicular and split the base in half, and the angles at B and C track each other exactly. Try to drag B to a position where ∠B ≠ ∠C — impossible. Equal sides force equal base angles.
How you build it
Construct an isosceles triangle with altitude.
- Draw the base BC using the segment tool.
- Construct the perpendicular bisector of BC — it crosses BC at its midpoint.
- Mark point A on the perpendicular line, above BC.
- Join A to B to draw one side of the triangle.
- Join A to C to complete the triangle.
The proof, step by step
Prove that the base angles of an isosceles triangle are equal.
- AB = AC (Given)
- AD = AD (Common side)
- ∠ADB = ∠ADC = 90° (Since AD ⊥ BC)
- ΔABD ≅ ΔACD (By RHS Congruence Rule)
- ∠B = ∠C (Corresponding Parts of Congruent Triangles)
Worked example
In ΔABC, AB = AC = 10 cm and BC = 12 cm. AD is drawn perpendicular to BC. Find the length of AD.
Since AB = AC, ΔABC is isosceles. The altitude AD to the base BC bisects it, so BD = DC = 6 cm. In right ΔABD, by Pythagoras theorem, AD² = AB² - BD² = 100 - 36 = 64. Thus, AD = 8 cm.
- 6 cm
- 8 cm — correct
- 10 cm
- 12 cm