The big picture
Eleven ideas about triangles fit on one screen.
| Idea | Remember | Example |
|---|---|---|
| Three lines | They shut in a triangle, unless two are parallel or all meet at one spot. | A parallel pair spoils it. |
| By sides | Three match: equilateral. Exactly two: isosceles. None: scalene. | 26, 26, 48 mm is isosceles. |
| By angles | The widest corner names it. | 85°, 60°, 35° is acute-angled. |
| The angle sum | The three corners add to 180°. | 45° + 55° + 80° = 180°. |
| The missing one | 180 minus the two you know. | 180 − 40 − 75 = 65°. |
| The exterior angle | It equals the two far corners added. | 45° + 55° = 100°. |
| Will it close | Each length is less than the other two added. | 6, 8, 10 closes. 2, 4, 7 does not. |
| Longest, widest | They face each other. | 100 mm is across from 90°. |
| Direct or roundabout | A road with a bend is longer than the direct road. So each side is shorter than the other two added. Checking the longest side is enough. A third side is longer than the difference of the other two. It is shorter than their sum. | 7 and 12: the third side is between 5 and 19. |
| Altitudes | An altitude is the perpendicular from a corner to the opposite side. Extend that side if it is needed. Its length is the height. Look at the two corners on that side. If either is obtuse, the foot falls outside. If either is 90°, the foot lands on that corner. If both are acute, it falls inside. In a right-angled triangle, two altitudes are the legs. | An obtuse triangle has two altitudes outside. |
| Two angles and a side | Take two angles at the ends of a side. A triangle exists when their sum is less than 180°. If the sum is 180° or more, it does not. The side length does not matter. The third angle is 180° minus their sum. | 38° and 64° leave 78°. |
See Part I, Chapter 7, A Tale of Three Intersecting Lines. Try its questions on triangles.