All propositions
Proposition I.15
Vertical angles
If two straight lines cut one another, they make the vertical angles equal.
When two lines cross, the opposite angles are equal. Euclid proves it from the fact that adjacent angles on a straight line sum to two rights — a pattern used constantly afterward.
Step 1 of 3
Two crossing lines
Given
Let the straight lines AB and CD cut one another at E. The vertical pairs are AED with CEB, and AEC with BED.