14. Two straight lines are said to be perpendicular to each other, when their meeting forms equal adjacent angles; thus the lines A B and CD are perpendicular to each other.

B

A

-D

Two adjacent angles, as CAB and BAD, have a common vertex, as A; and a common side, as A B.

15. A RIGHT ANGLE is one which

is formed by a straight line and a perpendicular to it; as the angle САВ.

C

Acute and obtuse angles have their sides oblique to each other, and are sometimes called oblique angles.

17. PARALLEL LINES are such as, being in the same plane, cannot meet, however far either way both of them may be produced; as the lines AB, CD.

18. When a straight line, as EF, intersects two parallel lines, as AB, CD, the angles formed A by the intersecting or secant line take particular names, thus:

INTERIOR ANGLES ON THE SAME C SIDE are those which lie within the parallels, and on the same

A

B

C

-D

E

G

B

D

H

F

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 EGB, 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 E GB, 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.

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.

22. An EQUILATERAL TRIANGLE is one which has its three sides equal; as the triangle ABC.

A

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 ABC and DEF, Art. 22. An OBTUSE-ANGLED TRIANGLE is one which has an obtuse angle; as the triangle GHI, 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

Α

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.