Proof.

Let any straight line AB be set up on the straight line CD. Now, if the angle CBA is equal to the angle ABD, then they are per definition two right angles and the proof is finished.

However, if angle CBA is not equal to the angle ABD, then let BE be drawn from the point B at right angles to CD (proposition I.11). Hence the angles DBE, CBE are two right angles.

The angle CBE is equal to the sum of two angles CBA and ABE:

Adding angle DBE to each side, we get

But we also have that angle DBA is equal to the sum of the two angles DBE and EBA:

Adding angle CBA to each side, we get

Since the right-hand-sides of (*) and (**) are equal, the left-hand-sides must also be equal, meaning that

But because angles CBE and DBE are two right angles, then angles DBA and CBA must also make up two right angles.

Drag e.g. ?.