221. The Greater Side's Secret · Why longer sides face bigger angles
In any triangle the larger angle always lies opposite the longer side. Drag C however you like — whichever of AB and AC is longer always faces the bigger angle, so the claim never breaks.
What this lesson covers
Try to break it
Drag C around. Side AB stays the same; AC and BC change. The larger angle always sits opposite the longer side. Drag C so AC > AB and the angle at B opens wider than the angle at C. Try to find a position where the longer side faces the smaller angle — impossible.
How you build it
Construct a triangle, making one side clearly longer than another.
- Place point A as the apex of the triangle.
- Place point B so that AB will be the longer side.
- Place point C so that AC is clearly shorter than AB.
- Draw segment AB, the longer side.
- Draw segment BC.
- Draw segment CA to complete triangle ABC.
The proof, step by step
Prove that the angle opposite the longer side is greater than the angle opposite the shorter side.
- In ΔACD, AC = AD (by construction).
- Therefore, ∠ACD = ∠ADC (angles opposite to equal sides).
- In ΔBDC, ∠ADC is an exterior angle, so ∠ADC > ∠B.
- From 2 and 3, ∠ACD > ∠B.
- Since ∠ACB contains ∠ACD, ∠ACB > ∠ACD > ∠B. Hence Proved.
Worked example
In ΔXYZ, XY = 12 cm and XZ = 9 cm. Which of the following is true regarding the angles?
Since XY > XZ (12 > 9), the angle opposite XY (∠Z) must be greater than the angle opposite XZ (∠Y). Thus, ∠Z > ∠Y.
- ∠Z > ∠Y — correct
- ∠Y > ∠Z
- ∠Z = ∠Y
- ∠X > ∠Z