Page:Euclid's Elements 1714 Barrow translation.djvu/30

18 b = AH. and join FC, and IC; and produce ACG.

Becaue CE c=EA, and EF c=EB, and the angle FEC d = BEA, the angle ECF e hall be equal to EAB. By the like argument is the angle ICH = ABH. Therefore the whole angle ACD (f BCG) g is greater than either the angle CAB or ABC. Which was to be demontrated.

Two angles of any triangle ABC, which way oever they be taken, are les than two right angles.

Let the ide BC be produced. Becaue the angle ACD + ACB a = 2 right angles, and the angle ACD b A, c therefore A + ACB  then two right angles. After the ame manner is the angle B + ACB then two right. Latly, the ide AB being produced, the angle A + B will be alo les than two right angles. Which was to he demontrated.

Coroll.

1. Hence it follows that in every triangle, wherein one angle is either right or obtue, the two others are acute angles.

2. If a right line AE make unequal angles with another right line D, one acute AED, the other obtue AEC, a perpendicular AD let fall from any point A to the other line CD, hall fall on that fide the acute is of.

For if AC, drawn on the ide of the obtue angle, be a perpendicular, then in the triangle AEC hall AEC + ACE be * greater than two right angles. Which is contrary to the prectdent Prop.

3. All the angles of an equilateral triangle, and the two angles of an Ioceles triangle that are upon the bae, are acute. PROP.