A Tale of Three Intersecting Lines
Two angles and the side between them
Construct a triangle from two angles and the side between them.
Now we're given two angles and the side between them (the included side). Construct ∆ABC with AB = 5 cm, ∠A = 45°, ∠B = 80°.
- Step 1: draw AB = 5 cm.
- Step 2: draw a 45° arm at A and an 80° arm at B, both going up.
- Step 3: where the arms cross is C.
But do the arms always cross? Not always.
Some cases are obvious: if both angles are 90° or more, the arms stand straight up or lean away from each other, and never meet.
A trickier case: ∠A = 40°. How big can ∠B be? If the arm from B leans too far, it never reaches the arm from A. The borderline is when the arm from B is parallel to the arm from A. Then AB is a transversal, and interior angles on the same side add to 180°: ∠B = 180° − 40° = 140°.
So with ∠A = 40°, there's no triangle when ∠B is 140° or more. And the length of AB played no part! The rule:
| Two angles add to | Result |
|---|---|
| less than 180° | a triangle ✓ |
| exactly 180° | the arms are parallel and never meet |
| more than 180° | the arms lean apart |
So 35° and 150° (185°) give no triangle; 70° and 30° (100°) do.
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