Proposition I-13

Proposition I-13. If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.
Proof.

We know that the angle CBE is equal to the two angles CBA and ABE. Add angle DBE to both.

Therefore the angles DBE and CBE equals the three angles CBA, ABE, and DBE.
(CN-2)

Let the line AB be set up on the straight line CD.

If angle CBA equals angle DBA, they are two right angles.
(Defn-10)

If CBA is not a right angle, let BE be drawn at right angles to DC.

Therefore, the angles DBE and CBE are right angles.

Next, angle DBA is equal to angle DBE and EBA. Add angle CBA to both.

This gives the angles DBA and CBA is equal to the three angles CBA, ABE, and DBE.
(CN-2)

Therefore, angles DBA and CBA are equal to the angles CDE and DBE, which is the sum of two right angles. (CN-1)

QED