PROPOSITION XXIV.

When two straight lines are cut by a third, if the alternate-interior angles are equal, the two straight lines are parallel.

Corollary I. When two straight lines are cut by a third, if a pair of corresponding angles are equal, the lines are parallel.

Corollary II. When two straight lines are cut by a third, if the sum of two interior angles on the same side of the secant line is equal to two right angles, the two lines are parallel.

PROPOSITION XXV.

If two parallel lines are cut by a third straight line, the alternateinterior angles are equal.

Corollary I. If two parallel lines are cut by a third straight line, any two corresponding angles are equal.

Corollary II. If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles.

PROPOSITION XXVI.

The sum of the three angles of any triangle is equal to two right angles. Corollary. If one side of a triangle is extended, the exterior angle is equal to the sum of the two interior opposite angles.

PROPOSITION XXVII.

The sum of all the angles of any polygon is equal to twice as many right angles, less four, as the figure has sides.

PROPOSITION XXVIII.

Two parallelograms are equal when two adjacent sides and the included angle of the one are equal to two adjacent sides and the included angle of the other.

Corollary. Two rectangles are equal when they have equal bases and equal altitudes.

PROPOSITION XXIX.

The opposite sides of a parallelogram are equal, and the opposite angles are equal.

PROPOSITION XXX.

If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

PROPOSITION XXXI.

If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

PROPOSITION XXXII.

The diagonals of a parallelogram bisect each other.

BOOK II.

THE CIRCLE.

1. DEFINITIONS. A circle is a portion of a plane bounded by a curve, all the points of which are equally distant from a point within it called the centre.

The curve which bounds the circle is

called its circumference.

Any straight line drawn from the centre to the circumference is called a radius.

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Any straight line drawn through the centre and terminated each way by the circumference is called a diameter.

In the figure, O is the centre, and the curve ABCEA is the circumference of the circle; the circle is the space included within the circumference; OA, OB, OC, are radii; AOC is a diameter.

By the definition of a circle, all its radii are equal; also all its diameters are equal, each being double the radius.

If one extremity, O, of a line OA is fixed, while the line revolves in a plane, the other extremity, A, will describe a circumference, whose radii are all equal to OA.

2. Definitions. An arc of a circle is any portion of its circumference; as DEF.

A chord is any straight line joining two points of the circumference; as DF. The arc DEF is said to be subtended by its chord DF.

Every chord subtends two arcs, which together make up the whole circumference. Thus, DF subtends both the arc DEF and the arc DCBAF. When an

arc and its chord are spoken of, the arc less than a semi-circumference, as DEF, is always understood, unless otherwise stated.

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A segment is a portion of the circle included between an arc and its chord; thus, by the segment DEF is meant the space included between the arc DF and its chord.

A sector is the space included between an arc and the two radii drawn to its extremities; as AOB.

3. From the definition of a circle it follows that every point within the circle is at a distance from the centre which is less than the radius; and every point without the circle is at a distance from the centre which is greater than the radius. Hence the locus of all the points in a plane which are at a given distance from a given point is the circumference of a circle described with the given point as a centre and with the given distance as a radius.

4. POSTULATE. A circumference may be described with any point as centre and any distance as radius.

PROPOSITION I.-THEOREM.

5. Two circles are equal when the radius of the one is equal to the radius of the other.

Let the second circle be superposed upon the first, so that its centre falls upon the centre of the first; then will the two circumferences coincide throughout.

For, if any point of either circumference falls outside of the other circle, the line joining it with the common centre must cross the circumference of that circle.

The whole line will be a radius of one circle, the portion of it within the other circle will be a radius of that other circle, and we shall have two unequal radii, which is contrary to our hypothesis.

PROPOSITION II.-THEOREM.

6. Every diameter bisects the circle and its circumference.

Let AMBN be a circle whose centre is O; then any diameter AOB bisects the circle and its circumference.

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For, if the figure ANB be turned about AB as an axis and superposed upon the figure AMB, the curve ANB will coincide with the curve AMB, since all the points of both are equally distant from the centre. (v. Proof of Proposition I.) The two figures then coincide throughout, and are therefore equal in all respects. Therefore AB divides both the circle and its circumference into equal parts.

7. Definitions. A segment equal to one-half the circle, as the segment AMB, is called a semicircle. An arc equal to half a circumference, as the arc AMB, is called a semi-circumference.