14. DEF. A Plane Figure is a figure, all points of which are in the same plane.

15. DEF. Geometry is the science which treats of position, magnitude, and form.

Points, lines, surfaces, and solids, with their relations, are the geometrical conceptions, and constitute the subject-matter of Geometry.

16. Plane Geometry treats of plane figures.

Plane figures are either rectilinear, curvilinear, or mixtilinear. Plane figures formed by straight lines are called rectilinear figures; those formed by curved lines are called curvilinear figures ; and those formed by straight and curved lines are called mixtilinear figures.

17. DEF. Figures which have the same form are called Similar Figures. Figures which have the same extent are called Equivalent Figures. Figures which have the same form and extent are called Equal Figures.

୦୦

ON STRAIGHT LINES.

18. If the direction of a straight line and a point in the line be known, the position of the line is known; that is, a straight line is determined in position if its direction and one of its points be known.

Hence, all straight lines which pass through the same point in the same direction coincide.

Between two points one, and but one, straight line can be drawn; that is, a straight line is determined in position if two of its points be known.

Of all lines between two points, the shortest is the straight line; and the straight line is called the distance between the two points.

The point from which a line is drawn is called its origin.

19. If a line, as CB, 4

B, be produced through C, the portions CB and CA may be regarded as different lines having opposite directions from the point C.

В

B, has two opposite

Hence, every straight line, as A B, A directions, namely from A toward B, which is expressed by saying line AB, and from B toward A, which is expressed by saying line B A.

A

B

с

20. If a straight line change its magnitude, it must become longer or shorter. Thus by prolonging A B to C, 4 AC AB+BC; and conversely, BC-AC-A B.

If a line increase so that it is prolonged by its own magnitude several times in succession, the line is multiplied, and the resulting line is called a multiple of the given line.

Thus, if AB÷

then AC 2 A B, A D:

=

=

It must also be possible to divide a given straight line into an assigned number of equal parts. For, assumed that the ath part of a given line were not attainable, then the double, triple, quadruple, of the nth part would not be attainable. Among these multiples, however, we should reach the nth multiple of this nth part, that is, the line itself. Hence, the line itself would not be attainable; which contradicts the hypothesis that we have the given line before us.

Therefore, it is always possible to add, subtract, multiply, and divide lines of given length.

21. Since every straight line has the property of direction, it must be true that two straight lines have either the same direction or different directions.

Two straight lines which have the same direction, without coinciding, can never meet; for if they could meet, then we should have two straight lines passing through the same point in the

22. Two straight lines which lie in the same plane and have different directions must meet if sufficiently prolonged; and must have one, and but one, point in common.

Conversely Two straight lines lying in the same plane which do not meet have the same direction; for if they had different directions they would meet, which is contrary to the hypothesis that they do not meet.

Two straight lines which meet have different directions; for if they had the same direction they would never meet (§ 21), which is contrary to the hypothesis that they do meet.

ON PLANE ANGLES.

23. DEF. An Angle is the difference in direction of two lines. The point in which the lines (prolonged if necessary) meet is called the Vertex, and the lines are called the Sides of the angle.

An angle is designated by placing a letter at its vertex, and one at each of its sides. In reading, we name the three letters, putting the letter at the vertex between the other two. When the point is the vertex of but one angle we usually name the letter at the vertex only; thus, in Fig. 1, we read the angle by

calling it angle A. But in Fig. 2, H is the common vertex of two angles, so that if we were to say the angle H, it would not be known whether we meant the angle marked 3 or that marked 4. We avoid all ambiguity by reading the former as the angle E HD, and the latter as the angle EHF.

The magnitude of an angle depends wholly upon the extent

of opening of its sides, and not upon their

length. Thus if the sides of the angle BA C,

B

namely, AB and AC, be prolonged, their extent of opening will not be altered, and the A size of the angle, consequently, will not be changed.

24. DEF. Adjacent Angles are angles having a common vertex and a common side between them. Thus the angles CDE and CDF are adjacent angles.

E

D

A

C

F

25. DEF. A Right Angle is an angle included between two straight lines which meet each other so that the two adjacent angles formed by producing one of the lines through the vertex are equal. Thus if the straight line AB meet the straight line C D so that the adjacent angles ABC and ABD are equal to one another, each of these angles is called a right angle.

26. DEF. Perpendicular Lines are lines which make a right angle with each other.

27. DEF. An Acute Angle is an angle less than a right angle; as the angle BA C.

28. DEF. An Obtuse Angle is an angle greater than a right angle; as the angle DEF.

C

D

B

B

F

C

D

E

29. DEF. Acute and obtuse angles, in distinction from right angles, are called oblique angles; and intersecting lines which are not perpendicular

to each other are called oblique lines.

30. DEF. The Complement of an angle is the difference between a right angle and the given angle. Thus ABD is the complement of the angle DB C; also DBC is the com

A

D

31. DEF. The Supplement of an angle is the difference between two right angles and the given angle. Thus ACD is the supplement of the angle D C B; also D C B is the supplement of the angle A C D.

32. DEF. Vertical Angles are angles which have the same vertex, and their sides extending in opposite directions. Thus the angles AOD and COB are vertical angles, as also the angles AO C and DO B.

ON ANGULAR MAGNITUDE.

33. Let the lines B B' and A A' be in the same plane, and let BB' be perpendicular to A A' at the point 0.

Suppose the straight line OC to move in this plane from coincidence with O A, about the point O as a pivot, to the position OC; then the line OC describes or generates the angle A O C

A

C

A'

The amount of rotation of the line, from the position 0 A to the position OC, is the Angular Magnitude A O C.

If the rotating line move from the position OA to the position O B, perpendicular to OA, it generates a right angle; to the position O A' it generates two right angles; to the position O B', as indicated by the dotted line, it generates three right angles; and if it continue its rotation to the position OA, whence it started, it generates four right angles.

Hence the whole angular magnitude about a point in a plane is equal to four right angles, and the angular magnitude about a point on one side of a straight line drawn through that point is equal to two right angles.