the line AD remaining in the same place; CB will, in this

new position EB, make angles EBA

and EBD equal to the former, and therefore all of them equal to each other. But the four angles ABC, CBD, DBE, and EBA constitute, a

-D

B

bout the point B, a complete revolu

tion; or the line BA in forming them,

by its successive openings, would return into its original place, --and consequently each of those angles is a right angle.

The angle contained by the opposite portions DA and DB

of a straight line is hence equal to two right angles; and, for the same reason, all the angles ADC, CDE, EDF and FDB, formed at the point D and on the same side of the straight line AB, are together equal to two right angles.

E

-B

5. The sides of a right angle are said to be perpendicular to each other.

6. An acute angle is less than a right

angle.

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7. An obtuse angle is greater than a right angle.

8. One side of an angle forms with the other produced a supplemental or exterior angle.

9. A vertical angle is formed by the production of both its sides.

10. The inverted divergence of the two sides of an angle, or the defect of the angle from four right angles, is named the reverse angle.

The angle DBE is vertical to ABC, ABD is the supplemental or exterior angle, and the angle made up of ABD, DBE, and EBC, or the opening formed by the regression of AB through the points D and E into the position BC, is the reverse angle.

-D

E

It is apparent that vertical angles, or those formed by the same lines in opposite directions, must be equal; for the angles CBA and BD which stand on the straight line CD, being equal to two right angles, are equal to ABD and DBE, and, omitting the common angle ABD, there remains CBA equal to DBE.

11. Two straight lines are said to be inclined to each other, if they meet when produced; and the angle so formed is called their inclination.

12. Straight lines which have no inclination, are termed parallel.

13. A figure is a plane surface included by a linear boundary called its perimeter.

14. Of rectilineal figures, the triangle is contained by three straight lines.

15. An isosceles triangle is that which has two of its sides equal.'

16. An equilateral triangle is that which has all its sides equal.

17. A triangle whose sides are unequal, is named scalene.

It will be shown (I. 9. cor.) that every triangle has at least two acute angles. The third angle may therefore, by its character, serve to discriminate a triangle.

18. A right-angled triangle is that which has a right angle.

19. An obtuse angled triangle is that

which has an obtuse angle.

20. An acute angled triangle is that

which has all its angles acute.

21. Any side of a triangle may be called its base, and the opposite angular point its vertex.

22. A quadrilateral figure is contained by four straight

lines.

23. Of quadrilateral figures, a trapezoid (1) has two parallel sides:

24. A trapezium (2) has two of its sides parallel, and the other two equal, though not parallel, to each other:

25. A rhomboid (3) has its opposite sides equal :

26. A rhombus (4) has all its sides equal:

27. An oblong, or rectangle, (5) has a right angle, and its opposite sides equal:

28. A square (6) has a right angle, and all its sides equal.

29. A quadrilateral figure, of which the opposite sides are parallel, is called a parallelogram.

30. The straight line which joins obliquely the opposite angular points of a quadrilateral figure, is named a diagonal.

31. If an angle of a rectilineal figure be less than two right angles, it protrudes, and is called salient; if it be greater than two right angles, it makes a sinuosity, and is termed re-entrant.

Thus the angle ABC is re-entrant, and the rest of the angles of the polygon ABCDEF are salient at A, C, D, E and F.

E

B

F

32. A rectilineal figure having more than four sides, bears the general name of a polygon.

33. A circle is a figure described by the revolution of a straight line about one of its extremities :

34. The fixed point is called the centre of the circle, the describing line its radius, and the boundary traced by the remote end of that line its circumference.

35. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

It is obvious that all radii of the same circle are equal to each other and to a semidiameter. It likewise appears, from the slightest inspection, that a circle can only have one centre, and that circles are equal which have equal diameters.

36. Figures are said to be equal, when, applied to each other, they wholly coincide; they are equivalent, if, without coinciding, they yet contain the same space.