right angles; then IH and CD will be parallel (Prop. XIX.), and hence we shall have two lines GB, GH, drawn through the same point G and parallel to CD, which is impossible (Ax. 12.): hence, GB and GH should coincide, and OGB+GOD is equal to two right angles. In the same manner it may be proved that OGA+GOC is equal to two right angles.

Cor. 1. If OGB is a right angle, GOD will be a right angle also: therefore, every straight line perpendicular to one of two parallels, is perpendicular to the other.

Cor. 2. If a straight line meet two parallel lines, the alternate angles will be equal.

C

A

E

D

B

Let AB, CD, be the parallels, and FE the secant line. The sum OGB+ GOD is equal to two right angles. But the sum OGB+OGA is also equal to two right angles (Prop. I.). Taking from each, the angle OGB, and there remains CGA GOD. In the same manner we may prove that GOC=OGB.

Cor. 3. If a straight line meet two parallel lines, the opposite exterior and interior angles will be equal. For, the sum OGB+GOD is equal to two right angles. But the sum OGB +EGB is also equal to two right angles. Taking from each the angle OGB, and there remains_GOD=EGB. In the same manner we may prove that AGE=GOC. '

Cor. 4. We see that of the eight angles formed by a line cutting two parallel lines obliquely, the four acute angles are equal to each other, and so also are the four obtuse angles.

PROPOSITION XXI. THEOREM.

If a straight line meet two other straight lines, making the sum of the interior angles on the same side less than two right angles, the two lines will meet if sufficiently produced.

Let the line EF meet the two lines CD, IH, making the sum of the interior angles OGH, A GOD, less than two right angles: then will IH and CD meet if sufficiently produced.

For, if they do not meet they are parallel (Def.12.). But they are not parallel, for if they were,

C

I

F

E

G

B

H

D

the sum of the interior angles OGH, GOD, would be equal to two right angles (Prop. XX.), whereas it is less by hypothesis: hence, the lines IH, CD, are not parallel, and will therefore meet if sufficiently produced.

Cor. It is evident that the two lines IH, CD, will meet on that side of EF on which the sum of the two angles OGH, GOD, is less than two right angles

PROPOSITION XXII. THEOREM.

Two straight lines which are parallel to a third line, are paralle to each other.

Let CD and AB be parallel to the third line EF; then are they parallel to each other.

Draw PQR perpendicular to EF, and cutting AB, CD. Since AB is parallel to EF, PR will be perpendicular to AB (Prop.T XX. Cor. 1.); and since CD is parallel to EF, PR will for a like reason be perpen-C dicular to CD. Hence AB and CD are. perpendicular to the same straight line; hence they are parallel (Prop. XVIII.).

.

A

R

T

Q

D

B

PROPOSITION XXIII. THEOREM.

Two parallels are every where equally distant.

perpendicular to CD (Prop. XX. Cor. 1.): and we are now to show that they will be equal to each other.

If GF be drawn, the angles GFE, FGH, considered in reference to the parallels AB, CD, will be alternate angles, and therefore equal to each other (Prop. XX. Cor. 2.). Also, the straight lines EG, FH, being perpendicular to the same straight line AB, are parallel (Prop. XVIII.); and the angles EGF, GFH, considered in reference to the parallels EG, FH, will be alternate angles, and therefore equal. Hence the two triangles EFG, FGH, have a common side, and two adjacent angles in each equal; hence these triangles are equal (Prop. VI.) ; therefore, the side EG, which measures the distance of the parallels AB and CD at the point E, is equal to the side FH, which measures the distance of the same parallels at the point F.

PROPOSITION XXIV. THEOREM.

If two angles have their sides parallel and lying in the same direction, the two angles will be equal.

Let BAC and DEF be the two angles, having AB parallel to ED, and AC to EF; then will the angles be equal.

For, produce DE, if necessary, till it meets AC in G. Then, since EF is parallel to GC, the angle DEF is equal to DGC (Prop. XX. Cor. 3.); and since

DG is parallel to AB, the angle DGC is equal to BAC; hence the angle DEF is equal to BAC (Ax. 1.).

Scholium. The restriction of this proposition to the case where the side EF lies in the same direction with AC, and ED in the same direction with AB, is necessary, because if FE were produced towards H, the angle DEH would have its sides parallel to those of the angle BAC, but would not be equal to it. In that case, DEH and BAC would be together equal to two right angles. For, DEH+DEF is equal to two right angles (Prop. I.); but DEF is equal to BAC: hence, DEH +BAČ is equal to two right angles.

PROPOSITION XXV. THEOREM.

In every triangle the sum of the three angles is equal to two right angles.

Let ABC be any triangle: then will the angle C+A+B be equal to two right angles.

B

AD

For, produce the side CA towards D, and at the point A, draw AE parallel to BC. Then, since AE, CB, are parallel, and CAD cuts them, the exterior angle DAE will be equal to its inte rior opposite one ACB (Prop. XX. Cor. 3.); in like manner, since AE, CB, are parallel, and AB cuts them, the alternate angles ABC, BAE, will be equal: hence the three angles of the triangle ABC make up the same sum as the three angles CAB, BAE, EAD; hence, the sum of the three angles is equal to two right angles (Prop. I.).

Cor. 1. Two angles of a triangle being given, or merely their sum, the third will be found by subtracting that sum from two right angles.

Cor. 2. If two angles of one triangle are respectively equal to two angles of another, the third angles will also be equa, and the two triangles will be mutually equiangular.

Cor. 3. In any triangle there can be but one right angle : for if there were two, the third angle must be nothing. Still less, can a triangle have more than one obtuse angle.

Cor. 4. In every right angled triangle, the sum of the two acute angles is equal to one right angle.

Cor. 5. Since every equilateral triangle is also equiangular (Prop. XI. Cor.), each of its angles will be equal to the third part of two right angles; so that, if the right angle is expressed by unity, the angle of an equilateral triangle will be expressed by 3.

Cor. 6. In every triangle ABC, the exterior angle BAD is equal to the sum of the two interior opposite angles B and C. For, AE being parallel to BC, the part BAE is equal to the angle B, and the other part DAE is equal to the angle C.

PROPOSITION XXVI. THEOREM.

The sum of all the interior angles of a polygon, is equal to two right angles, taken as many times less two, as the figure has sides.

Let ABCDEFG be the proposed polygon. If from the vertex of any one angle A, diagonals B AC, AD, AE, AF, be drawn to the vertices of all the opposite angles, it is plain that the polygon will be divided into five triangles, if it has seven sides; into six triangles, if it has eight; and, in general, into as many triangles, less two, as the polygon has sides; for, these triangles may be considered, as having the point A for a common vertex, and for bases, the several sides of the polygon, excepting the two sides which form the angle A. It is evident, also, that the sum of all the angles in these triangles does not differ from the sum of all the angles in the polygon: hence the sum of all the angles of the polygon is equal to two right angles, taken as many times as there are triangles in the figure; in other words, as there are units in the number of sides diminished by two.

Cor. 1. The sum of the angles in a quadrilateral is equal to two right angles multiplied by 4-2, which amounts to four

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 2x (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 convex 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.

G

A

α

F

B/b

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