Triangles
99. The Isosceles Balance · Equal sides guarantee equal base angles
The base angles ∠ABC and ∠ACB are always equal.
In an isosceles triangle, the two base angles (the angles opposite the equal sides) are always equal. This follows from the symmetry of the equal sides and is the cornerstone of many proofs.
What this lesson covers
Try to break it
Can you make the base angles unequal? Try dragging A! (Remember, the construction keeps AB equal to AC).
How you build it
Construct an isosceles triangle.
- Draw the base segment BC.
- Construct the perpendicular bisector of BC.
- Mark a point A on the perpendicular bisector.
- Draw the side from A to B.
- Draw the side from A to C to complete the isosceles triangle.
The proof, step by step
Prove that the base angles ∠ABC and ∠ACB of the isosceles triangle are equal.
- Draw the altitude AM from A to BC, meeting BC at M.
- In right triangles ABM and ACM: AB = AC (Given), AM = AM (Common side), ∠AMB = ∠AMC = 90° (Since AM ⟂ BC).
- Therefore, ΔABM ≅ ΔACM by RHS Congruence Rule.
- Hence, ∠ABM = ∠ACM (Corresponding Parts of Congruent Triangles).
Worked example
In an isosceles triangle ABC, AB = AC. If the vertex angle ∠A measures 40°, what is the measure of each base angle?
Sum of angles = 180°. Base angles are equal. 2x + 40 = 180 => 2x = 140 => x = 70°.
- 40°
- 70° — correct
- 80°
- 100°