Cor. 1. From this proposition, and proposition XXVIII. Cor. 1., it follows, that if one angle of a rhomboid be right, all the angles will be right.

Cor. 2. Therefore in the rectangle and square (see Definitions) all the angles are right, and in the latter all the sides are equal.

Cor. 3. The diagonal divides a rhomboid into two equivalent triangles.

Cor. 4. Parallels included between two other parallels are equal.

Conversely, if the opposite sides of a quadrilateral be equal, or if the opposite angles be equal, the figure will be a rhomboid.

In the quadrilateral ABCD let the opposite sides be equal, the figure will be a rhomboid.

Let the diagonal AC be drawn, then, the triangles ABC, ADC, are equal, since the three sides of the one are respectively equal to those of the other, (Prop. XIII.); therefore the angles BAC, DCA, opposite the equal sides BC, DA,

B

D

arc equal; therefore DC is parallel to AB; the angles ACB, CAD, opposite the equal sides AB, DC, are also equal; BC is therefore parallel to AD (Prop. XXII. Cor. 1.): hence ABCD is a rhomboid.

Next, let the opposite angles be equal.

Then the sum of the angles BAD, ADC, must be equal to the sum of the angles DCB, CBA; therefore each sum is equal to two right angles (Prop. XXVIII. Cor. 1.); therefore AB, DC, are parallel (Prop. XXII. Cor. 1.) For similar reasons AD, BC, are parallel; therefore the figure is a rhom boid.

PROPOSITION XXXIV. THEOREM.

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

In the quadrilateral ABCD (preceding diagram), let AB be equal and parallel to DC, then will ABCD be a rhomboid. For the diagonal AC makes the alternate angles BAC, DCA, equal (Prop. XXIII. Cor. 2.); so that in the triangles

ABC, CDA, two sides, and the included angle in each, are respectively equal; these triangles are, therefore, equal, (Prop. VIII.): the angle ACB is, therefore, equal to the angle CAD; hence AD is parallel to BC (Prop. XXII. Cor. 1.), and the other two sides are parallel by hypothesis; therefore ABCD is a rhomboid.

Cor. If, in addition, the parallel sides be each equal to a third side, the rhomboid will be either a rhombus or a square, according as it has, or has not, a right angle.

Scholium. It has been proved (Prop. VIII.) that two triangles are equal, when two sides and the included angle in the one, are respectively equal to two sides and the included angle in the other; we may now infer further, that two triangles are equivalent or equal in surface, when two sides of the one are respectively equal to two sides of the other, and the sum of the included angles equal to two right angles. For, let the triangles ADC, BCD, having two sides, AD, DC, in the one equal to the two BC, CD, D in the other, be placed as in the margin, a side of the one coinciding with the equal side in the other; let also the included angles ADC, BCD, be together equal to two right angles, and let AB, BD be drawn.

A

Then, since the angles ADC, BCD, are together equal to two right angles, the lines AD, BC, are parallel (Prop. XXII. Cor. 1.), but they are also equal by hypothesis; hence, by the above proposition, the figure ABCD is a rhomboid; now, the triangle ADC is half the rhomboid (Prop. XXXII. Cor. 3.), so also is the triangle BCD; these triangles are, therefore, equivalent.

PROPOSITION XXXV. THEOREM.

The diagonals of a rhomboid bisect each other.

The diagonals AC, BD, of the rhomboid ABCD (see preceding diagram) are mutually bisected in the point P.

For, since AB, CD, are parallel, the angles PAB, PBA, are respectively equal to the angles PCD, PDC (Prop. XXIII. Cor. 2.), and AB being also equal to CD, the triangles PAB, PCD, are equal (Prop. IX.) therefore the sides AP, CP, opposite the equal angles ABP, CDP, are equal, as also the sides BP, DP, opposite the other equal angles. The diagonals of a rhomboid, therefore, bisect each other.

PROPOSITION XXXI. THEOREM.

Conversely, if the diagonals of a quadrilateral bisect each other, the figure is a rhomboid.

If the diagonals AC, BD, bisect each other, ABCD is a rhomboid.

For, the two sides AP, BP, and included an-D gle, being equal to the two sides CP, PD, and included angle, the side AB is equal to the side CD (Prop. VIII. Cor. 2.) For similar reasons AD is equal to CB; hence (Prop. XXXIII.) the quadrilateral is a rhomboid.

P

C

B

It may be remarked that in this and the four succeeding books of these elements, the lines concerned in each proposition are supposed to be all situated in the same plane.

THE CIRCLE AND THE MEASUREMENT OF ANGLES.

DEFINITIONS.

1. Every line which is not a right line, or composed of right lines, is a curve line.

2. A circle is the surface terminated by a curve line, which is in every part equally distant from a point within, called

the centre.

3. The boundary of a circle is called its circumference.

4. The radius is a right line drawn from the centre of a circle to the circumference, and it follows from Def. 2. that all radii are equal.

5. The diameter of a circle is a right line drawn through the centre. Hence, according to Def. 4, it is twice the radius.

In the annexed diagram the surface enclosed by the boundary curve line, is the circle; the curve line AFHGBDE, the circumference; the lines CE, CD, drawn A from the centre to the circumference, are radi; and the right line AB drawn through the centre C, terminating in the circumference, the diameter.

6. An are is any portion of a circumference.

H

G

7. A chord of an arc is a right line joining the extremities of that arc. Hence, it is said to subtend the arc.

8. A segment of a circle is the surface included between an arc and its chord.

The surface FHG, included between the arc FHG and the chord FG is a segment; so likewise is the surface included by the same chord and the arc FAEDBG.

9. A sector of a circle is the surface included between two radii, and the intercepted arc. The surface ECD is a sector of a circ e.

10. A tangent is a line which touches A a circumference, so that if it is produced, it will not cut it. The line AB is a tan

gent.

11. A secant is a line that cuts a circle lying partly within and partly without. The line CD is a secant.

C

12. One circle is said to touch another, when they have one point common, and but one.

D

B

13. A line is inscribed in a circle, when its extremities are in the circumference, as AB or BC.

14. An angle is inscribed in a circle, when its sides are inscribed.

The angle ABC is an inscribed angle.

B

15. An inscribed triangle is one which, like BAC, has its three angular points in the circumference.

16. A polygon is inscribed in a circle, when its sides are inscribed; and under the same circumstances, a circle is said to circumscribe a polygon.

17. A circle is inscribed in a polygon, when all its sides touch the circumference, and the polygon is said to circumscribe the circle.

18. By an angle in the segment of a circle, is meant one which has its vertex in the arc, and whose sides intercept the chord And by an angle at the centre, is meant one which has its vertex in the centre. In both cases, the angles are said to be subtended by the chords, or the arcs, which their sides include.

PROPOSITION I. THEOREM.

A diameter divides a circle and its circumference into tw● equal parts.

Let AEDF be a circle, and AB a diameter. Now, if the figure AEB be applied to AFB, their common base AB retaining its position, the curve line AEB must fall ex- A actly on the curve line AFB, otherwise there would, in the one or the other, be points unequally distant from the centre, which is contrary to the definition of a circle.

PROPOSITION II. THEOREM.

Every chord is less than the diameter.

Let AD (see preceding diagram) be any chord. Draw the radii CA, CD, to its extremities.

We shall then have AD<

AC+CD (Prop. X. B. II.); or AD<AB.

Cor. Hence, the greatest line which can be inscribed in a circle is its diameter.