that is to say, they are in different directions, and are, therefore, not parallel; that is to say, lines which meet at any distance however great, are not parallel, and therefore, conversely, parallel lines

never meet.

XIII.

If a straight line fall on two parallel lines, it makes the alternate angles equal to each other, and the two interior angles equal to two right angles.

Let FBD, CE be parallel straight lines (Fig. 8); A B C a straight line cutting them in B and C respectively; the angle FBC shall be equal to BCE, and the angles D B C and D C B shall be together equal to two right angles.

Because BD and CE are parallel, the angle ACE is equal to the angle A B D, and, therefore, to the vertical angle F B C.

To each of these add the angle D BC. Then the angles ACE, D B C are together equal to the two F B C, D B C, that is. to two right angles.

XIV.-EUCLID I. 32.

If one side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three angles of every triangle are together equal to two right angles.

Let A B C (Fig. 9) be a triangle, and let one of its sides, BC, be produced to D; the exterior

angle A CD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles ABC, BCA, CA B, are together equal to two right angles.

Through the point C draw CE parallel to AB; and because A B is parallel to C E, and A C meets them, the alternate angles BAC, ACE are equal. Again, because A B is parallel to C E, and BD falls upon them, the exterior angle E C D is equal to the interior and opposite angle A B C. Therefore the whole angle A CD, consisting of the two angles A CE, ECD, is equal to the two interior and opposite angles A B C, BA C.

To each of these add the angle AC B. Then the three interior angles of the triangle ABC, BAC, BCA will be together equal to the two, ACB, ACD; that is, to two right angles.

Cor. If from any point in a straight line making an oblique angle with a second line a third straight line be drawn perpendicular to the second, the perpendicular will fall on the same side of the oblique line with the acute angle, and on the opposite side to the obtuse angle.

XV.-EUCLID I. 32, Cor.

All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

For any rectilinear figure ABCDE (Fig. 10)

can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of the angles. And by the last proposition all the angles of these triangles are equal to twice as many right angles as there are triangles or sides to the figure. And the same angles are equal to the internal angles of the figure, together with the angles at the point F, which is the common vertex of the triangles; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

XVI.

If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles equal, the third sides are also equal, and the remaining angles of the one to the remaining angles of the other, viz. those to which the equal sides are opposite.

Let ABC, DEF (Fig. 11) be two triangles, having the sides AB, AC, equal to DE, DF, respectively, and the angle B A C equal to EDF. The third side BC will also be equal to E F, and the angles ABC, ACB to the angles DE F, DFE respectively.

Let the triangle ABC be superimposed on DEF, so that the point A shall lie on D, A B

upon DE, and the plane A B C upon the plane DEF.

Then, because A B is equal to DE, the point B will coincide with E; and because the angle BAC is equal to EDF, and AB coincides with DE, the line B C will lie on the line DF; and because A C is equal to D F, point C will coincide with D, and therefore line DC with D F, and the triangles will wholly coincide. Therefore the angle ABC is equal to D E F, and angle ACB to DFE.

Cor.-In triangles having two sides of the one equal to two sides of the other respectively, if the bases are not equal, the angles at the vertex are unequal.

XVII.

If two triangles have two angles and one side of the one equal to two angles and the corresponding side of the other, the triangles are altogether equal; viz. the remaining sides of the one to the remaining sides of the other to which the equal angles are opposite.

If two angles of the one triangle are equal to two angles of the other, the third angles are also equal (Prop. 14).

Let the triangles ABC, DEF (Fig. 11), have the angles BA C, B C A, equal to EDF, EF D, respectively, and therefore the third angle ABC equal to DE F, and let any side A C be equal to

D

DF. The remaining sides A B, B C shall also be equal to E D, E F respectively.

Let the triangle ABC be superimposed on DEF, so that AC shall coincide with D F. Then, because the angle BAC is equal to E D F, the line AB will lie on the line DE, and for a like reason the line C B will lie on FE. And because B is a point in line A B, it will fall somewhere in line DE, and because it is a point in line C B it will fall somewhere in F E. It will therefore fall on the point E in which DE and FE intersect each other, and the sides A B, BC will coincide with the sides DE, FE respectively. Therefore the sides A B, BC are equal to DE, FE, respectively, and the triangles are equal in every respect.

XVIII.

The angles at the base of an isosceles triangle, or a triangle having two equal sides, are equal to each other; and, conversely, if the angles at the base of a triangle are equal, the opposite sides are also equal to each other.

Let A B C (Fig. 12) be a triangle in which A B is equal to A C. The angle ABC will be equal to A CB.

Let the triangle A B C be taken up and laid on its face, so that the side A B which was on the left of the triangle shall now be on the right, and let A'C'B' be the triangle in the reversed position.