ALTERNATE INTERIOR ANGLES lie within the parallels, and on different sides of the secant line, but are not adjacent to each other; as the angles BGH, GH C, and also AGH, GHD.

ALTERNATE EXTERIOR ANGLES lie without the parallels, and on different sides of the secant line, but not adjacent to each other; as the angles EG B, CHF, and also the angles A GE, DHF.

OPPOSITE EXTERIOR and INTERIOR ANGLES lie on the same side of the secant line, the one without and the other within the parallels, but not adjacent to each other; as the angles EGB, GHD, and also EGA, GHC, are, respectively, the opposite exterior and interior angles.

PLANE FIGURES.

19. A PLANE FIGURE is a plane terminated on all sides by straight lines or curves.

The boundary of any figure is called its perimeter.

20. When the boundary lines are straight, the space they enclose is called a RECTILINEAL FIGURE, or POLYGON; as the figure ABCDE.

E

A

D

B

21. A polygon of three sides is called a TRIANGLE; one of four sides, a QUADRILATERAL; one of five, a PENTAGON; one of six, a HEXAGON; one of seven, a HEPTAGON; one

of eight, an OCTAGON; one of nine, a NONAGON; one of ten, a DECAGON; one of eleven, an UNDECAGON; one of twelve, a DODECAGON; and so on.

The side opposite to the right angle is called the hypothenuse; as the side J L.

24. An ACUTE-ANGLED TRIANGLE is one which has three acute angles; as the triangles A B C and DEF, Art. 22. An OBTUSE-ANGLED TRIANGLE is one which has an obtuse angle; as the triangle G H I, Art. 22.

Acute-angled and obtuse-angled triangles are also called oblique-angled triangles.

25. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel.

26. A RECTANGLE is any parallelogram whose angles are right angles; as the parallelogram A B C D.

D

C

A

B

30. A BASE of a polygon is the side on which the polygon is supposed to stand. But in the case of the isosceles triangle, it is usual to consider that side the base which is not equal to either of the other sides.

31. An equilateral polygon is one which has all its sides equal. An equiangular polygon is one which has

all its angles equal. A regular polygon is one which is equilateral and equiangular.

32. Two polygons are mutually equilateral, when all the sides of the one equal the corresponding sides of the other, each to each, and are placed in the same order.

Two polygons are mutually equiangular, when all the angles of the one equal the corresponding angles of the other, each to each, and are placed in the same order.

33. The corresponding equal sides, or equal angles, of polygons mutually equilateral, or mutually equiangular, are called homologous sides or angles.

AXIOMS.

34. An AXIOM is a self-evident truth; such as,

1. Things which are equal to the same thing, are equal to each other.

2. If equals be added to equals, the sums will be equal. 3. If equals be taken from equals, the remainders will be equal.

4. If equals be added to unequals, the sums will be unequal.

5. If equals be taken from unequals, the remainders will be unequal.

6. Things which are double of the same thing, or of equal things, are equal to each other.

7. Things which are halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. The whole is equal to the sum of all its parts.

10. A straight line is the shortest line that can be drawn from one point to another.

11. From one point to another only one straight line can be drawn.

12. Through the same point only one parallel to a straight line can be drawn.

13. All right angles are equal to one another.

14. Magnitudes which coincide throughout their whole extent, are equal.

POSTULATES.

35. A POSTULATE is a self-evident problem; such as, 1. That a straight line may be drawn from one point to another.

2. That a straight line may be produced to any length. 3. That a straight line may be drawn through a given point parallel to another straight line.

4. That a perpendicular to a given straight line may be drawn from a point either within or without the line.

5. That an angle may be described equal to any given angle.

PROPOSITIONS.

36. A DEMONSTRATION is a course of reasoning by which a truth becomes evident.

37. A PROPOSITION is something proposed to be demonstrated, or to be performed.

A proposition is said to be the converse of another, when the conclusion of the first is used as the supposition in the second.

38. A THEOREM is something to be demonstrated.

39. A PROBLEM is something to be performed.

40. A LEMMA is a proposition preparatory to the demonstration or solution of a succeeding proposition.

41. A COROLLARY is an obvious consequence deduced from one or more propositions.

42. A SCHOLIUM is a remark made upon one or more preceding propositions.

43. An HYPOTHESIS is a supposition, made either in the