to two angles (7) *; therefore the sum of all the angles at the point B is equal to four right angles.

THEOREM II.

10. If at a point in a straight line two other straight lines upon opposite sides of it make the sum of the adjacent angles equal to two right angles, these two lines form a straight line.

Let the straight line DB meet the two lines, AB, BC, so as to make ABD+DBC= two right angles: then AB and BC form a straight line.

B

D

E
C

For if A B and BC do not form a straight line, draw BE SO

that A B and BE shall form a straight line; then

ABD+DBE = two right angles (7);

but by hypothesis,

therefore

ABD+DBC = two right angles;

DBE=DBC

the part equal to the whole, which is absurd (Axiom 6); therefore A B and BC form a straight line.

THEOREM III.

11. If two straight lines cut each other, the vertical angles are

equal.

Let the two lines, AB, CD, cut each other at E; then AEC DEB.

THEOREM IV.

12. Two angles whose sides have the same or opposite directions are equal.

1st. Let BA and BC, including the angle B, have respectively the same direction as ED and EF, including the angle E; B4 then angle B = angle E.

For since BA has the same direction as

C

A

F

ED, and BC the same as EF, the difference of direction of BA and BC must be

the same as the difference of direction of ED and EF; that is,

13. Definition. Parallel Lines are such as

have the same direction; as AB and CD.

14. Corollary. Parallel lines can never meet. For, since parallel lines have the same direction, if they coincided at one point, they would coincide throughout and form one and the same straight line.

Conversely, straight lines in the same plane that never meet, however far produced, are parallel. For if they never meet they cannot be approaching in either direction, that is, they must have the same direction.

15. Axiom. Two lines parallel to a third are parallel to each other.

16. Definition. When parallel lines are cut by a third, the

secant, or cutting line, are called internal angles on the same side; as AGH and GHC, or BGH and GHD. Two internal angles on opposite sides of the secant, and not adjacent, are called alternate internal angles; as AGH and GHD, or BGH and GHC.

Two angles, one external, one internal, on the same side of the secant, and not adjacent, are called opposite external and internal angles; as EGA and GHC, or EGB and GHD.

THEOREM V.

17. If a straight line cut two parallel lines,

1st. The opposite external and internal angles are equal.

2d. The alternate internal angles are equal.

3d. The internal angles on the same side are supplements of

2d. The alternate internal angles, AGH and GHD, or BGH and GHC, are equal, since their sides have opposite directions (12).

3d. The internal angles on the same side, AGH and GHC, or BGH and GHD, are supplements of each other; for AGH is the supplement of AGE (8), which has just been proved equal to GIIC. In the same way it may be proved that BGH and GHD are supplements of each other.

THEOREM VI.

CONVERSE OF THEOREM V.

18. If a straight line cut two other straight lines in the same plane, these two lines are parallel,

1st. If the opposite external and internal angles are equal.

2d. If the alternate internal angles are equal.

3d. If the internal angles on the same side are supplements of each other.

F

For, if through the point Ga line is drawn parallel to CD, it will make the opposite external and internal angles equal, and the alternate internal angles equal, and the internal angles on the same side equal (17); therefore it must coincide with AB; that is, A B is parallel to CD.

PLANE FIGURES.

DEFINITIONS.

19. A Plane Figure is a portion of a plane bounded by lines either straight or curved.

When the bounding lines are straight, the figure is a polygon, and the sum of the bounding lines is the perimeter.

20. An Equilateral Polygon is one whose sides are equal each to each.

21. An Equiangular Polygon is one whose angles are equal each to each.

22. Polygons whose sides are respectively equal are mutually equilateral.

23. Polygons whose angles are respectively equal are mutually equiangular.

Two equal sides, or two equal angles, one in each polygon, similarly situated, are called homologous sides, or angles.

24. Equal Polygons are those which, being applied to each other, exactly coincide.

25. Of Polygons, the simplest has three sides, and is called a triangle; one of four sides is called a quadrilateral; one of five, a pentagon; one of six, a hexagon; one of eight, an octagon ; one of ten, a decagon.

TRIANGLES.

26. A Scalene Triangle is one which has

no two of its sides equal; as ABC.

27. An Isosceles Triangle is one which has two of its sides equal; as DEF.

28. An Equilateral Triangle is one whose sides are all equal; as IGH.