Page:Euclid's Elements 1714 Barrow translation.djvu/28

16 If the angles ABC,ABD be equal, a then they make two right angles; if unequal, then from the point B b let there be erected a perpendicular BE. Becaue the angle ABC c = to a right + ABE, and the angles ABD d = to a right &minus; ABE, therefore all be ABC &minus; ABD e = to two right angles + ABE &minus; ABE = two right angles. Which was to he demonrated.

1. Hence, if one angle A B D be right, the other ABC is alo right; if one acute, the other is obtue, and o on the contrary. 2. If more right lines than one and upon the ame right line at the ame point, the angles {{lsh}all be equal to two right. 3. Two right lines cutting each other make angles equal to four right. 4. All the angles made about one point, make four right; as appears by Coroll. 3.

If to any right line AB, and a point therein B, two right lines, not drawn from the ame de, do make the angles ABC, ABD on each de equal to two right, the lines CB,BD all make one rait line.

If you deny it, let CB, BE make one right line, then all be the angle ABC + ABE a = two right angles b = ABC + ABD. Which is aburd.

If two right lines AB, CD cut thro' one another, then are the two angles which are oppote, viz. CEB, AED, equal one to the other.

For the. angle AEC + CEB a = to two right angles = AEC + AED; b therefore CEB = AED. Which was to be demonrated.