VERTICAL ANGLES. PROPOSITION XI. If two straight lines cut one another, the vertical (or opposite) angles shall be equal to one another. K Let the two straight lines DEL, FEK cut one another. Then shall DEF be equal to the vertical KEL. For if the figure were taken up, reversed, and placed so that each of the arms of ▲ DEK might fall along the former position of the other arm; that is, so that ED might fall along the former position of EK, and EK along that of ED; then would EK produced, or EF, fall along that of ED produced, or EL. Thus EF and ED would fall along the former positions of EL and EK. Therefore DEF is equal to ▲ KEL. Wherefore if two straight lines &c. Q.E.D. RIGHT ANGLES. DEFINITION. When one straight line, standing on another straight line, makes the adjacent angles equal to one another, each of them is called a right angle; and the straight line which stands upon the other is said to be perpendicular to it. EUCLID'S AXIOM.-All right angles are equal to one another. PROPOSITION XII. To draw a perpendicular to a given straight line from a given point in the same. and A the given point in it. It is required to draw from A a perpendicular to BC. From AB, AC cut off equal parts AD, AE. With D and E as centres, describe circles with equal radii intersecting in F. Join AF Then shall AF be perpendicular to BC. Join DF and EF. Because the side AFis common to the As DAF, EAF, and AD is equal to AE, and also DF is equal to EF; therefore As DAF, EAFare equal in all respects, (I. 7) and thus DAF is equal to ▲ EAF. Therefore AF is perpendicular to BC. (Def.) Q.E.F. PROPOSITION XIII. To draw a perpendicular to a given straight line from a given point without it. B Let A be the given point without the given straight line BC. It is required to draw from A a perpendicular to BC. With centre A describe a circle cutting BC at D and E. With centres D and E describe circles with equal radii intersecting in F Join AF, cutting BC in G. Then shall AG be perpendicular to BC. Join AD, AE, FD, and FE. Now because in the As DAF, EAF the sides DA, AF, and FD are respectively equal to EA, AF, and FE; ..▲ DAF is equal to ▲ EAF. (I. 7) Again, because in the As DAG, EAG the sides DA, AG and the included angle DAG are respectively equal to EA, AG and the included angle EAG; ..LDGA is equal to ▲ EGA; .. AG is perpendicular to BC. Wherefore from a given point &c. (I. 4) (Def.) Q.E. F. PROPOSITION XIV. The angles which one straight line makes with another upon one side of it are either two right angles or are together equal to two right angles. B Let the straight line AB make with CD upon one side of it the s ABC, ABD. These shall either be two right angles or shall together be equal to two right angles. If the ABC is equal to the ▲ ABD; then each of them is a right angle. (Def.) But if not, through B draw BE perpendicular to CD; (I. 12) then the ABC is equal to the 4s CBE and EBA together; therefore the LS ABC and ABD are together equal to the LS CBE, EBA, and ABD, of which CBE is a right angle, and S EBA and ABD together make up a right angle. Therefore the LS ABC and ABD are together equal to two right angles. Wherefore the angles which one straight line &c. Q.E.D. COROLLARY.-If two angles be equal to one another, then the adjacent angles, formed by producing an arm of each through the vertex, will be equal to one another. PARALLELS. DEFINITION. Straight lines, lying in the same plane and which will never meet, though produced ever so far both ways, are called parallel. PLAYFAIR'S AXIOM.-Two straight lines which cut one another cannot be parallel to the same straight line. Straight lines which are parallel to the same straight line are parallel to one another. A B Let the straight lines A and B be each of them parallel to C. Then shall A and B be parallel to one another. For if not, they will meet if produced, and there will thus be two intersecting straight lines parallel to the same straight line, which is impossible. Wherefore straight lines &c. B (Axiom.) Q.E.D. |