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26.

A scalene triangle is that which has three unequal sides.

27.

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

28.

An obtuse-angled triangle is that which has an obtuse angle.

29.

An acute-angled triangle is that which has three acute angles.

30.

Of quadrilateral or four-sided figures, a square has all its sides equal, and all its angles right angles.

31.

An oblong is that which has all its angles right angles, but has not all its sides equal.

32.

A rhombus has all its sides equal, but its angles are not right angles.

33.

A rhomboid has its opposite sides equal to each other, but all its sides are not equal, nor its angles right angles.

34.

All other four-sided figures besides these, are called Trapeziums.

35.

Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.

A.

A parallelogram is a four-sided figure, of which the opposite sides are parallel; and the diameter, or the diagonal, is the straight line joining two of its opposite angles.

POSTULATES.

1.

LET it be granted that a straight line may be drawn from any one point to any other point.

2.

That a terminated straight line may be produced to any length in a straight line.

3.

And 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, are equal to one another. 7. Things which are halves of the same, are equal to one another.

8.

Magnitudes which coincide with one another-that is, which exactly fill the same space-are equal to one another.

9.

The whole is greater than its part.

10.

Two straight lines cannot inclose a space.

11.

All right angles are equal to one another.

12.

If a straight line meets two straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.

EXPLANATION OF TERMS.

1. A Proposition in geometry, as its name implies, is something proposed either to be done or demonstrated.

2. Propositions fall into two classes, problems and theorems..

3. A Problem proposes some geometrical construction to be done -e.g., the construction of a figure.

4. A Theorem proposes some geometrical property to be demonstrated.

5. A Postulate is a problem so simple that it is unnecessary to point out the method of doing it.

6. An Axiom is a theorem, the truth of which is self-evident.

7. A Corollary is an inference made immediately from the discussion of the proposition to which it is subjoined.

PROPOSITION 1.-Problem.

To describe an equilateral triangle upon a given finite straight line.

Let AB be the given straight line.

It is required to describe an equilateral triangle upon AB.

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B

Construction. From the centre A, at the distance AB, describe the circle BCD (Postulate 3); from the centre B, at the distance BA, describe the circle ACE; and from C, one of the points in which the circles cut one another, draw the straight lines CA, CB to the points A, B (Post. 1).

Then ABC shall be an equilateral triangle.

Proof. Because the point A is the centre of the circle BCD, r 1. AC is equal to AB (Definition 15);

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and because the point B is the centre of the circle ACE,

2. BC is equal to BA.

But it has been proved that AC is equal to AB; therefore AC, BC are each of them equal to AB; but things which are equal to the same thing are equal to one another (Axiom 1); therefore

3. AC is equal to BC.

Wherefore AB, BC, CA are equal to one another; and therefore 4. The triangle ABC is equilateral;

and it is described upon the given straight line AB.

Which was required to be done.

PROPOSITION 2.-Problem.

From a given point, to draw a straight line equal to a given straight

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