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right angles: hence, if all the angles of a quadrilateral are equal, each of them will be a right angle; a conclusion which sanctions the seventeenth Definition, where the four angles of a quadrilateral are asserted to be right angles, in the case of the rectangle and the square.
Cor. 2. The sum of the angles of a pentagon is equal to two right angles multiplied by 5-2, which amounts to six right angles: hence, when a pentagon is equiangular, each angle is equal to the fifth part of six right angles, or to g of one right angle.
Cor. 3. The sum of the angles of a hexagon is equal to 2× (6-2,) or eight right angles; hence in the equiangular hexagon, each angle is the sixth part of eight right angles, or of one.
Scholium. When this proposition is applied to polygons which have re-entrant angles, each reentrant angle must be regarded as greater than two right angles. But to avoid all ambiguity, we shall henceforth limit our reasoning to polygons with salient angles, which might otherwise be named conver polygons. Every convex polygon is such that a straight line, drawn at pleasure, cannot meet the contour of the polygon in more than two points.
PROPOSITION XXVII. THEOREM.
If the sides of any polygon be produced out, in the same direction, the sum of the exterior angles will be equal to four right angles.
Let the sides of the polygon ABCDFG, be produced, in the same direction; then will the sum of the exterior angles a+b+c+d+f+g, be equal to four right angles.
For, each interior angle, plus its exterior angle, as A+a, is equal to two right angles (Prop. I.). But there are as many exterior as interior angles, and as many of each as there are sides of the polygon: hence, the sum of all the interior and exterior angles is equal to twice as many right angles as the polygon has sides. Again, the sum of all the interior angles is equal to two right angles, taken as many times, less two, as the polygon has sides (Prop. XXVI.); that is, equal to twice as many right angles as the figure has sides, wanting four right angles. Hence, the interior angles plus four right
angles, is equal to twice as many right angles as the polygon has sides, and consequently, equal to the sum of the interior angles plus the exterior angles. Taking from each the sum of the interior angles, and there remains the exterior angles, equal to four right angles.
In every parallelogram, the opposite sides and angles are equal.
Let ABCD be a parallelogram: then will AB=DC, AD=BC, A=C, and ADC=ABC.
PROPOSITION XXVIII. THEOREM.
For, draw the diagonal BD. The triangles ABD, DBC, have a common side BD; and since AD, BC, are parallel, they have also the angle ADB DBC, (Prop. XX. Cor. 2.); and since AB, CD, are parallel, the angle ABD BDC: hence the two triangles are equal (Prop. VI.); therefore the side AB, opposite the angle ADB, is equal to the side DC, opposite the equal angle DBC; and the third sides AD, BC, are equal: hence the opposite sides of a parallelogram are equal.
Again, since the triangles are equal, it follows that the angle A is equal to the angle C; and also that the angle ADC composed of the two ADB, BDC, is equal to ABC, composed of the two equal angles DBC, ABD: hence the opposite angles of a parallelogram are also equal.
Cor. Two parallels AB, CD, included between two other parallels AD, BC, are equal; and the diagonal DB divides the parallelogram into two equal triangles.
PROPOSITION XXIX. THEOREM.
If the opposite sides of a quadrilateral are equal, each to each, the equal sides will be parallel, and the figure will be a parallelogram.
Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz. AB DC, and AD=BC; then will these sides be parallel, and the figure be a parallelogram.
For, having drawn the diagonal BD, the triangles ABD, BDC, have all the sides of the one equal to
the corresponding sides of the other; therefore they are equal, and the angle ADB, opposite the side AB, is equal to DBC, opposite CD (Prop. X.); therefore, the side AD is parallel to BC (Prop. XIX. Cor. 1.). For a like reason AB is parallel to CD therefore the quadrilateral ABCD is a parallelogram.
PROPOSITION XXX. THEOREM.
If two opposite sides of a quadrilateral are equal and parallel, the remaining sides will also be equal and parallel, and the figure will be a parallelogram.
Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel; then will the figure be a parallelogram.
For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles ABD, BDC, are equal (Prop. XX. Cor. 2.); moreover, the side DB is common, and the side AB=DC; hence the triangle ABD is equal to the triangle DBC (Prop. V.); therefore, the side AD is equal to BC, the angle ADB DBC, and consequently AD is parallel to BC; hence the figure ABCD is a parallelogram.
PROPOSITION XXXI. THEOREM.
Let ABCD bé a parallelogram, AC and B DB its diagonals, intersecting at E, then will AE EC, and DE=EB.
The two diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other.
Comparing the triangles ADE, CEB, we find the side AD=CB (Prop. XXVIII.), the angle ADE=CBE, and the angle DAE ECB (Prop. XX. Cor. 2.); hence those triangles are equal (Prop. VI.); hence, AE, the side opposite the angle ADE, is equal to EC, opposite EBC; hence also DE is equal to EB.
Scholium. In the case of the rhombus, the sides AB, BC, being equal, the triangles AEB, EBC, have all the sides of the one equal to the corresponding sides of the other, and are therefore equal: whence it follows that the angles AEB, BEC, are equal, and therefore, that the two diagonals of a rhombus cut each other at right angles.
OF RATIOS AND PROPORTIONS.
1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if A and B represent quantities of the same kind, the ratio of A to B is expressed by
The ratios of magnitudes may be expressed by numbers, either exactly or approximatively; and in the latter case, the approximation may be brought nearer to the true ratio than any assignable difference.
Thus, of two magnitudes, one of them may be considered to be divided into some number of equal parts, each of the same kind as the whole, and one of those parts being considered as an unit of measure, the inagnitude may be expressed by the number of units it contains. If the other magnitude contain a certain number of those units, it also may be expressed by the number of its units, and the two quantities are then said to be commensurable.
If the second magnitude do not contain the measuring unit an exact number of times, there may perhaps be a smaller unit which will be contained an exact number of times in each of the magnitudes. But if there is no unit of an assignable value, which shall be cortained an exact number of times in each of the magnitudes, the magnitudes are said to be incommensurable.
It is plain, however, that the unit of measure, repeated as many times as it is contained in the second magnitude, would always differ from the second magnitude by a quantity less than the unit of measure, since the remainder is always less than the divisor. Now, since the unit of measure may be made as small as we please, it follows, that magnitudes may be represented by numbers to any degree of exactness, or they will differ from their numerical representatives by less than any assignable quantity.
Therefore, of two magnitudes, A and B, we may conceive A to be divided into M number of units, each equal to A': then A = M × A': let B be divided into N number of equal units, each equal to A'; then B-Nx A'; M and N being integral numbers. Now the ratio of A to B, will be the same as the ratio of M x A' to Nx A'; that is the same as the ratio of M to N, since A' is a common unit.
In the same manner, the ratio of any other two magnitudes C and D may be expressed by PxC' to Q× C', P and Q being also integral numbers, and their ratio will be the same as that of P to Q.
2. If there be four magnitudes A, B, C, and D, having such B. D values that is equal to- then A is said to have the same ratio
to B, that C has to D, or the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, they are said to be in proportion.
To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C: D, and read, A is to B as C is to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms, the two means.
3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order.
4. Three quantities are in proportion, when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two.
5. Magnitudes are said to be in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents.
6. Magnitudes are in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent.
7. Magnitudes are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent.
8. Magnitudes are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent.
9. Equimultiples of two quantities are the products which arise from multiplying the quantities by the same number: thus, m × A, m× B, are equimultiples of A and B, the common multiplier being m.
10. Two quantities A and B are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. In such case, either of them is equal to a constant quantity divided by the other, and their product is constant.