PROP. XVII. THEOR. 18. 1 Eu. The greater side of every triangle is opposite

to, or subtends the greater angle. If ABC be a A of which the side AC > side AB, then ABC > L ACB.

PROP. XVIII. THEOR. 19. 1 Eu. The greater angle of every triangle is subtended by the greater side; or has the greater side opposite to it.

PROP. XIX. THEOR. 20. 1 Eu. Any two sides of a triangle are together

greater than the third side.
Let ABC be a A, then

AB + AC> BC,
AB + BC> AO
BC + AC AB

B

PROP. XX. THEOR. 21. I Eu. If from the ends of the side of a triangle, there

be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

Let ABC be a A ; from B, C, the ends of the side BC, draw BD, CD, from the point D within the A; then BD+DC<BA+AC, and 2 BDC > _BAC.

PROP. XXI. PROB. 22. 1 Eu.

To make a triangle of which the sides shall be

equal to three given straight lines, but any two whatever of these must be greater than the third. (19 Prop.)

Let A, B, C, be the 3 given straight lines. It is required to make a A, of which the sides shall be equal to A, B, C, respectively.

From D take DE unlimited towards E. Prop. 3.

Make DF= A,

FG = B,

GH = C. Post. 3. From cent. F, and dist. FD, descr. O DKL. From cent. G, and dist. GH, descr. O HLK.

Join KF and KG.

PROP. XXII. PROB. 23. 1 Eu.

At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle.

Let AB be the given str. line, A a given pt. in it, DCE the given L; to make an L at A in AB = _DCE. In CD, CE, take any points D, E.

Join DE.