or 180 degrees; and those on both sides of the line, from. the same point, will be equal to four right angles, or 360 degrees.

6. In every triangle, if one side be continued in the same direction, the outward angle will be as much above 90 degrees, as the adjoining inward angle is less than 90 degrees, and equal to both the other inward angles, which will more clearly appear, by inspecting the following figure, and that in position 3d. than by long demon

strations.

7. In a right angled triangle, if the base and perpendicular be equal, and hypothenuse be bisected at right angles, and the bisecting line continued, it will divide the right angle into two equal angles, each 45 degrees, and the acute angles at A and C, are each 45 degrees, and. the four angles make 180 degrees.

8. In a right angled triangle, as DEF, if the perpendicular be longer than the base, and the hypothenuse be bisected at right angles, the bisecting line will intersect the perpendicular, so that the upper end (cut off by this bisecting line) will exactly reach from the place of intersection, as at G, to the extreme end of the base, as to D; thus the lines DG and FG, are equal in length, and because the base DE, and perpendicular FE, are unequal in length, the acute angles at D and F are unequal, and the greatest angle will always be opposite the longest side. And as DG and FG are equal, the angles GDF and FDG are equal, by position 7th.

10. If the sides of a triangle be unequal in length, as GHI, the angles will be unequal in quantity, and the greatest angle will be opposite the longest side, and the least angle opposite the shortest side, as in the following figure; the three angles taken together, making two right angles, or 180 degrees.

30

86

52

G

H

11. If two sides of a triangle be equal in length, as the sides DE and FE, in the following figure, are equal, then the angles opposite these equal sides, will be equal, viz. the angles at D and F, are equal to each other.

D

GEOMETRICAL PROBLEMS.

PROBLEM I.

TO draw a line parallel to a given line AB, at any distance, as at C.

RULE.

Take, with a pair of compasses, the nearest distance between the point C, and the line AB, and with that distance, and one foot of the compasses in the line AB, as at A, describe an arch, as at D; then from the point C, draw a line to touch the arch at D, and it is done.

To bisect, or divide a line into two equal parts.

RULE.

With any distance in the compasses, greater than half the given line AB, and one foot of the compasses in A,

D