Parallel and Intersecting Lines
Linear pairs and vertically opposite angles
Learn about linear pairs and vertically opposite angles, and find angles without measuring.
Lines l and m cross, making angles a, b, c, d. Suppose a = 120°. Can you find the rest without measuring?
Yes! a and b together make a straight angle, which is 180°. So b = 180° − 120° = 60°. b and c also make a straight angle, so c = 120°. c and d too, so d = 60°.
Look: the angles facing each other came out equal (a = c = 120°, b = d = 60°). Is that always true?
Here is why it must be true, for any size of a. Straight angles are 180°, so a + b = 180° and a + d = 180°. Both make 180° with a, so b = d. In the same way, a + b = 180° and c + b = 180°, so a = c.
Names to remember:
| Name | Where | Rule |
|---|---|---|
| linear pair | two angles side by side on a straight line | add up to 180° |
| vertically opposite angles | two angles facing each other at the crossing | always equal |
Notice what we did: we didn't measure anything. We reasoned from a fact we already knew and reached an answer that must be true every time. That kind of argument is called a proof, and it is stronger than any number of measurements.
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