Constructions
142. The Isosceles Blueprint · Base and angles, meet at the peak
The base angles remain equal, so the triangle stays isosceles.
To construct an isosceles triangle with given base and base angle θ: draw the base AB, then draw rays from A and B making angle θ with AB on the same side. The rays meet at C. Both base angles equal θ, so the triangle is isosceles.
What this lesson covers
Try to break it
Drag the orange handle around A to change the base angle. The angle at B mirrors it, so AC = BC at every position. Try to set the handle so AC ≠ BC — you can't. The construction is locked symmetric.
How you build it
Construct an isosceles triangle from base and equal sides.
- Mark point A — one end of the base segment.
- Mark point B — the other end of the base segment.
- Draw segment AB — the base of the isosceles triangle.
- Place compass at A. Set radius equal to the given equal side AC. Draw an arc above AB.
- Keep the compass set to the same radius. Place it at B and draw an arc. Where the two arcs cross is point C.
- Mark point C exactly where the two arcs cross.
- Join A to C — the first equal side.
- Join B to C — the second equal side. Triangle ABC is complete!
The proof, step by step
Prove that the constructed triangle is isosceles.
- We are given that ∠A = ∠B.
- In any triangle, sides opposite equal angles are equal.
- Therefore, AC = BC.
- By definition, ΔABC is isosceles.
Worked example
In ΔABC, AB = 8 cm and ∠A = ∠B = 65°. Find the measure of ∠C.
The sum of angles in a triangle is 180°. ∠C = 180° - (65° + 65°) = 50°.
- 40°
- 50° — correct
- 65°
- 130°