Book 1 - Proposition 13

If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

Let any straight line AB standing on the straight line CD make the angles angle CBA and angle ABD.

I say that either the angles angle CBA and angle ABD are two right angles or their sum equals two right angles.

Now, if the angle CBA equals the angle ABD, then they are two right angles.

But, if not, draw BE from the point B at right angles to CD. Therefore the angles angle CBE and angle EBD are two right angles.

Since the angle angle CBE equals the sum of the two angles angle CBA and angle ABE, add the angle EBD to each, therefore the sum of the angles angle CBE and angle EBD equals the sum of the three angles angle CBA, angle ABE, and angle EBD.

Again, since the angle DBA equals the sum of the two angles angle DBE and angle EBA, add the angle ABC to each, therefore the sum of the angles angle DBA and angle ABC equals the sum of the three angles angle DBE, angle EBA, and angle ABC.

But the sum of the angles angle CBE and angle EBD was also proved equal to the sum of the same three angles, and things which equal the same thing also equal one another, therefore the sum of the angles angle CBE and angle EBD also equals the sum of the angles angle DBA and angle ABC. But the angles angle CBE and angle EBD are two right angles, therefore the sum of the angles angle DBA and angle ABC also equals two right angles.

Therefore if a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.
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