If the angles ABC,ABD be equal, aa def. 10.
b 11. 1.
c 19. ax.
d 3. ax.
e 2. ax. then they make two right angles; if unequal, then from the point B b let there be erected a perpendicular BE. Becauſe the angle ABC c = to a right + ABE, and the angles ABD d = to a right − ABE, therefore ſhall be ABC − ABD e = to two right angles + ABE − ABE = two right angles. Which was to he demonſtrated.
Corollaries.
1. Hence, if one angle A B D be right, the other ABC is alſo right; if one acute, the other is obtuſe, and ſo on the contrary. 2. If more right lines than one ſtand upon the ſame right line at the ſame 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.
PROP. XIV.
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If to any right line AB, and a point therein B, two right lines, not drawn from the ſame ſide, do make the angles ABC, ABD on each ſide equal to two right, the lines CB,BD ſhall make one ſtrait line.
If you deny it, let CB, BE make one right line, then ſhall be the angle ABC + ABE aa 13. 1.
b hyp.
c 9. ax. = two right angles b = ABC + ABD. Which is abſurd.
PROP. XV.
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If two right lines AB, CD cut thro' one another, then are the two angles which are oppoſite, viz. CEB, AED, equal one to the other.
For the. angle AEC + CEB aa 13. 1.b 3. ax. = to two right angles = AEC + AED; b therefore CEB = AED. Which was to be demonſtrated.
