Triangles
31. The Isosceles Balance · equal sides always hide equal angles
The base angles of an isosceles triangle are always equal.
An isosceles triangle has at least two equal sides. The angles opposite the equal sides — called the base angles — are also equal. Drag the apex; both base angles change together but always remain equal to each other.
What this lesson covers
Try to break it
Try to make ∠Q and ∠R different. No matter how you move P, they stay equal! This is the magic of isosceles triangles.
How you build it
Construct an isosceles triangle.
- Place point A (apex of the isosceles triangle).
- Open the Compass to the desired equal-side length: click A (centre), then click anywhere to set the radius. The arc shows every possible position for B and C.
- Pick point B on the arc (one of the base vertices).
- Pick point C elsewhere on the same arc (the other base vertex — AB = AC because they're both radii).
- Draw side AB.
- Draw side AC.
- Draw base BC. Two sides equal — the angles opposite them are equal too.
The proof, step by step
Prove that the base angles of an isosceles triangle are equal.
- Draw the altitude PM from P to QR, meeting QR at M.
- In triangles PMQ and PMR: PQ = PR (given), PM = PM (common), ∠PMQ = ∠PMR = 90°.
- By RHS congruence criterion, ΔPMQ ≅ ΔPMR.
- Therefore, ∠PQM = ∠PRM, which means ∠Q = ∠R.
Worked example
In an isosceles triangle PQR with PQ = PR, if ∠Q = 55°, find the measure of ∠P.
Since PQ = PR, ∠Q = ∠R = 55°. The sum of angles in a triangle is 180°, so ∠P = 180° - (55° + 55°) = 70°.
- 55°
- 60°
- 70° — correct
- 110°