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PROP. XXXI. THEOR. 32. 1 Eu.
If a side of any triangle be produced, the ex
terior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Let ABC be a A, and the side BC be prod. to D. Then shall the ext. 2 ACD = 2 CAB + L ABC, the two int. and oppo. Zs. And the 3 int. Ls, viz., _ ABC + _ BCA + Z CAB= 2 rt. Zs.
For any rectilin. figure ABCDE can be divided into as many as as the figure has sides, by drawing str. lines from a pt. F within the figure to each of its angles. Then, by the preceding proposition, all the ls of these As are equal to twice as many rt. Ls as there are As, i.e. as there are sides of the figure : and the same Zs are equal to the Ls of the figure, together with the Ls at the pt. F, which is the common vertex of the As; that is, together with 4 rt. Ls.
Prop. 14. Cor. 2.-All the ext. Ls of any rectilin. figure are together equal to 4 rt. Zs.
PROP. XXXII. THEOR. 33. 1 Eu.
The straight lines which join the extremities
of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.
Let AB, CD, be = and || str. lines, joined towards the same parts by the str. lines AC, BD : then shall AC and BD be = and | to each other.
PROP. XXXIII. THEOR. 34. 1 Eu. The opposite sides and angles of parallelo
grams are equal to one another, and the diameter bisects them, that is, divides them into two equal parts. N.B.-A parallelogram is a four-sided figure, of which the opposite sides are parallel ; and the diameter is the straight line joining two of its opposite angles.
Let ACDB be a om, of which BC is a diam. the opp. sides and Ls of the figure are = to one another; and BC bisects the figure.