10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

11. An obtuse angle is an angle which is greater than a right angle.

12. An acute angle is an angle which is less than a right angle.

13. A term or boundary is the extremity of anything.

14. A figure is that which is contained by one boundaries.

15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal.

16. This point is called the centre of the circle.

17. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference.

18. A semicircle is a plane figure contained by a diameter of a circle and the part of the circumference that it cuts off.

or more

D

The complement of a given angle is an angle which with the given angle makes up a right angle. The complement of an angle of 60° is an angle of 30°, because 60° + 30° = 90°.

The supplement of a given angle is an angle which with the given angle makes up two right angles. The supplement of an angle of 60° is an angle of 120°, because 60° + 120° = 180°.

11. No angle in Euclid is larger than two right angles.

14. The perimeter of a figure is the total length of the lines which bound it.

15. A radius of a circle is a straight line drawn from the centre to the circumference.

EXERCISES.

1. Derive and explain the words adjacent and vertical, as applied to (1) lines and (2) angles.

2. In the figure to the note on Def. 10 name all the adjacent and all the opposite vertical angles.

3. What are the complements of the following angles: 30°, 45°, 90°?

4. What are the supplements of the following angles: 30°, 45°, 90° ?

5. What kind of angle is the supplement of an acute angle ?

6. In what sense is the diagram in the note on Def. 10 a figure, and in what sense is it not a figure?

7. If the three sides of a triangle are 2, 3, and 4 feet long, what is the perimeter of the triangle? What is the semi-perimeter ?

19. A segment of a circle is a plane figure contained by a straight line and the part of the circumference that it cuts off.

20. Rectilineal figures are figures which are contained by straight lines.

21. A triangle is a plane figure contained by three straight lines.

22. A quadrilateral is a plane figure contained by four straight lines.

23. A polygon is a plane figure contained by more than four straight lines.

24. An equilateral triangle is a triangle which has three equal sides.

25. An isosceles triangle is a triangle which has only two sides equal.

26. A scalene triangle is a triangle which has three unequal sides.

27. A right-angled triangle is a triangle which has a right angle.

28. An cbtuse-angled triangle is a triangle which has an obtuse angle.

29. An acute-angled triangle is a triangle which has three acute angles.

30. Parallel straight lines are such as, being in the same plane, do not meet however far they are produced in either direction.

31. A parallelogram is a four-sided figure which has its opposite sides parallel.

32. A square is a four-sided figure which has all its sides equal and all its angles right angles.

33. A rhombus is a four-sided figure which has all its sides equal but its angles are not right angles.

34. A trapezium is a four-sided figure which has only two sides parallel.

24-29. There are seven kinds of triangle, namely, (1) equilateral, (2) right-angled isosceles, (3) obtuse-angled isosceles, (4) acute-angled isosceles, (5) right-angled scalene, (6) obtuse-angled scalene, and (7) acute-angled scalene.

27. In a right-angled triangle the side which subtends or is opposite to the right angle is called the hypotenuse.

27--29. In every triangle there are two acute angles.

One side of a triangle is often called the base, and the angle opposite the base is called the vertex. In an isosceles triangle the base is the side which is not equal to either of the other sides.

31. A diagonal of a quadrilateral is a straight line joining two of the opposite angles. A diagonal of a parallelogram is often called a diameter of the parallelogram.

An oblong is a four-sided figure which has all its angles right angles, but all its sides are not equal.

A rhomboid is a four-sided figure which has its opposite sides equal, but all its sides are not equal nor its angles right angles.

POSTULATES.

Let it be granted

1. That a straight line may be drawn from any one point to any other point.

2. That a finite, that is to say a terminated, straight line may be produced to any length in a straight line.

3. That a circle may be described from any centre, at any distance from that centre.

AXIOMS.

1. Things which are equal to the same thing are equal to one another.

2. If equals be added to equals the wholes are equal.

3. If equals be taken from equals the remainders are equal. 4. If equals be added to unequals the wholes are unequal.

5. If equals be taken from unequals the remainders are unequal.

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

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

8. Magnitudes which coincide with one another are equal.

9. The whole is greater than its part.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

12. If a straight line, falling on two other straight lines, makes

the two interior angles on the same side of it

together less than two right angles, these two straight lines will meet, if continually produced, on that side on which the angles are together less than two right angles.