Parallel and Intersecting Lines
Why the protractor disagrees
Find out why a protractor sometimes seems to disagree with the rules.
You may have noticed something odd. When you measure a linear pair, the two angles sometimes add to 179° or 181°, not exactly 180°. Vertically opposite angles sometimes differ by a degree. But we proved they must be equal! What's going on?
The proof is fine. There are two reasons for the difference:
- Measuring mistakes: lining the protractor up on the corner, reading a mark by eye. Each step can be a little off.
- Thick lines: a line in geometry has no thickness at all, but a pencil line is a thin strip. Where you measure inside that strip changes the reading slightly.
In geometry we study ideal lines and shapes, perfect versions of the ones we draw. For them we know by reasoning that a straight angle is 180°, so if a line splits it into two parts, the parts add to 180°.
When we measure real drawings, the answer comes out very close to what we reasoned. That closeness is why geometry is used so much in physics, art, engineering and architecture: it describes the real world well enough to build with.
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