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DAC, BCA, are likewise equal; hence the two angles BAC, DAC, are together equal to the two angles DCA, BCA; that is, the opposite angles BAD, DCB, are equal. Again, since the angles BAC, BCA, and the interjacent side of the triangle ABC are respectively equal to the angles DCA, DAC, and the interjacent side of the triangle CDA, the triangles are equal (Prop. XI.); therefore the side AB is equal to the side CD, the side BC to DA, and the angle B to the angle D; hence, in a rhomboid, the opposite sides and angles are equal.

Cor. 1. From this proposition, and Cor. 3. to Prop. XVII., 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.

PROPOSITION XXVIII. THEOREM.

XXVII.)

(Converse of Prop.

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.

A

B

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. XXV.); therefore the angles BAC, DCA, opposite the equal sides BC, DA, are equal; therefore DC is parallel to AB; the angles ACB, CAD, opposite the equal sides BC, DA, are also equal; BC is, therefore, parallel to AD (Prop. XII.); 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. XVII Cor. 3.); therefore AB, DC, are parallel (Prop. XII. Cor. 3.). For similar reasons AB, BC, are parallel; therefore the figure is a rhom

boid.

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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. XIV.); 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. XII.), 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.

P

For, let the triangles ADC, BCD, having two sides AĎ, DC, in the one equal to the two BC, CD, 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.

B

Then, since the angles ADC, BCD, are together equal to two right angles, the lines AD, BC, are parallel (Prop. XII. Cor. 3.), 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. XXVII. Cor. 3.), so also is the triangle BCD; these triangles are, therefore, equivalent.

PROPOSITION XXX. THEOREM.

The diagonals of a rhomboid bisect each other.

The diagonals AC, BD, of the rhomboid ABCD 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. XIV.), and AB being also equal to CD, the triangles PAB, PCD, are equal (Prop.

D

P

XI.); 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. (Converse of Prop. XXX.)

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

If the diagonals AC, BD (preceding diagram), bisect each other, ABCD is a rhomboid.

For the two sides AP, PB, and included angle being equal to the two sides CP, PD, and included angle, the side AB is equal to the side CD (Prop. VIII.). For similar reasons AD is equal to CB; hence (Prop. XXVIII.) the quadrilateral is a rhomboid.

BOOK II.

DEFINITIONS.

1. The altitude of a triangle is the distance of one of its sides, taken as a base, from the vertex of the opposite angle.

The perpendicular AD from the vertex A to the base BC, is the altitude of the triangle ABC.

B

D

2. The altitude of a rhomboid is the distance of one of its sides, considered as a base, from the opposite side.

3. The altitude of a trapezium is the

distance between its parallel sides.

4. A rectangle is said to be contained by its adjacent sides.

The rectangle ABCD is contained by the sides DA, AB. For brevity it is often referred to as the rectangle of DA, AB.

A

D

5. If, within a rhomboid, two straight lines parallel to the adjacent sides be drawn so as to intersect the diagonal in the same point; then, of the four rhomboids into which the figure is divided, those two through which the diagonal passes are said to be about the diagonal, and the other two are called their complements.

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In referring to a rhomboid it will be sufficient to employ the letters placed at two opposite corners.

PROPOSITION I. THEOREM.

The complements of the rhomboids about the diagonal of a rhomboid are equivalent.

Thus, in the above diagram, the rhomboids AG, GC, are equivalent.

The triangle ABD is equal to the triangle CDB; the triangle HGD to the triangle IDG, and the triangle EBG to the triangle FGB (Prop. XXVII. Cor. 3.); take the triangles HGĎ, EBG, from the triangle ABD, and there will remain the rhomboid AG; take, in like manner, from the other half of the rhomboid AC the triangles IDG, FGB, equal to the former two, and there will remain the rhomboid GC; these rhomboids, therefore, are equivalent.

PROPOSITION II. THEOREM.

Rhomboids are equal which have two sides, and the included angle in each equal.

Let the sides AB, AD, and the angle A in the rhomboid AC be respectively equal to the sides EF, EH, and the angle E in the rhomboid EG; these rhomboids are equal.

For the opposite sides

of rhomboids being equal, it follows that the four sides of the rhomboid AC

are respectively equal to

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those of the rhomboid EG; therefore, since the angles A and C are also equal, the two rhomboids are equal (Prop. XXV. Cor. B. I.)

Cor. 1. If a rhomboid and a triangle have two sides, and the included angle in the one respectively equal to two sides and

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