7. When one straight line stands on another so as to make the adjacent angles equal to one another, each of the angles is called a right angle; and each line is said to be perpendicular to the other.

AXIOMS. (i) If O is a point in a straight line AB, then a line OC, which turns about O from the position OA to the position OB, must pass through one position, and only one, in which it is perpendicular to AB.

(ii) All right angles are equal.

A right angle is divided into 90 equal parts called degrees (°); each degree into 60 equal parts called minutes ('); each minute into 60 equal parts called seconds (").

In the above figure, if OC revolves about O from the position OA into the position OB, it turns through two right angles, or 180°.

If OC makes a complete revolution about O, starting from OA and returning to its original position, it turns through four right angles, or 360°.

8. An angle which is less than one right angle is said to be acute.

That is, an acute angle is less than 90°.

9. An angle which is greater B than one right angle, but less than two right angles, is said to be obtuse.

That is, an obtuse angle lies between 90° and 180°.

B

A

A

10. If one arm OB of an angle turns until it makes a straight line with the other arm OA, the angle so formed is B called a straight angle.

A straight angle = 2 right angles=180°.

11. An angle which is greater than two right angles, but less than four right angles, is said to be reflex.

That is, a reflex angle lies between B 180° and 360°.

A

NOTE. When two straight lines meet, two angles are formed, one greater, and one less than two right angles. The first arises by supposing OB to have revolved from the position OA the longer way round, marked (i); the other by supposing OB to have revolved the shorter way round, marked (ii). Unless the contrary is stated, the angle between two straight lines will be considered to be that which is less than two right angles.

12. Any portion of a plane surface bounded by one or more lines is called a plane figure.

13. A circle is a plane figure contained by a line traced out by a point which moves so that its distance from a certain fixed point is always the same.

Here the point P moves so that its distance P from the fixed point O is always the same.

P

The fixed point is called the centre, and the bounding line is called the circumference.

14. A radius of a circle is a straight line drawn from the centre to the circumference. It follows that all radii of a circle are equal.

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

16. An are of a circle is any part of the circumference.

17. A semi-circle is the figure bounded by a diameter of a circle and the part of the circumference cut off by the diameter.

18. To bisect means to divide into two equal parts.

AXIOMS. (i) If a point ○ moves from A to B along the straight line Ā AB, it must pass through one position in which it divides AB into two equal parts. That is to say:

Every finite straight line has a point of bisection.

(ii) If a line OP, revolving about turns from OA to OB, it must pass through one position in which it divides the angle AOB into two equal parts.

That is to say:

Every angle may be supposed to have a line of bisection.

HYPOTHETICAL CONSTRUCTIONS.

B

B

A

From the Axioms attached to Definitions 7 and 18, it follows that we may suppose

(i) A straight line to be drawn perpendicular to a given straight line from any point in it.

(ii) A finite straight line to be bisected at a point. (iii) An angle to be bisected by a line.

SUPERPOSITION AND EQUALITY.

AXIOM. Magnitudes which can be made to coincide with one another are equal.

This axiom implies that any line, angle, or figure, may be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison, and it states that two such magnitudes are equal when one can be exactly placed over the other without overlapping.

This process is called superposition, and the first magnitude is said to be applied to the other.

POSTULATES.

In order to draw geometrical figures certain instruments are required. These are, for the purposes of this book, (i) a straight ruler (ii) a pair of compasses. The following Postulates (or requests) claim the use of these instruments, and assume that with their help the processes mentioned below may be duly performed.

Let it be granted:

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

2. That a FINITE (or terminated) straight line may be PRODUCED (that is, prolonged) to any length in that straight line.

3. That a circle may be drawn with any point as centre and with a radius of any length.

NOTES. (i) Postulate 3, as stated above, implies that we may adjust the compasses to the length of any straight line PQ, and with a radius of this length draw a circle with any point O as centre. That is to say, the compasses may be used to transfer distances from one part of a diagram to another.

(ii) Hence from AB, the greater of two straight lines, we may cut off a part equal to PQ the less.

For if with centre A, and radius equal to PQ, we draw an arc of a circle cutting AB at X, it is obvious that AX is equal to PQ.

A

X

B

P

INTRODUCTORY.

1. Plane geometry deals with the properties of such lines and figures as may be drawn on a plane surface.

2. The subject is divided into a number of separate discussions, called propositions.

Propositions are of two kinds, Theorems and Problems.

A Theorem proposes to prove the truth of some geometrical

statement.

A Problem proposes to perform some geometrical construction, such as to draw some particular line, or to construct some required figure.

3. A Proposition consists of the following parts:

The General Enunciation, the Particular Enunciation, the Construction, and the Proof.

(i) The General Enunciation is a preliminary statement, describing in general terms the purpose of the proposition.

(ii) The Particular Enunciation repeats in special terms the statement already made, and refers it to a diagram, which enables the reader to follow the reasoning more easily.

(iii) The Construction then directs the drawing of such straight lines and circles as may be required to effect the purpose of a problem, or to prove the truth of a theorem.

(iv) The Proof shews that the object proposed in a problem has been accomplished, or that the property stated in a theorem is true.

4. The letters Q.E.D. are appended to a theorem, and stand for Quod erat Demonstrandum, which was to be proved.