Cancel Matter Inserted.] [The proofs of Theorems 1, 2, 3 of Book I. having been found unsuitable for beginners, the following Modifications have been adopted in accordance with a Resolution of the Association.] THEOR. I. All straight angles are equal to one another. Let AB, AC be the arms of a straight angle, whose vertex is A, and DE, DF the arms of another straight angle, whose vertex is D: Ax. 2a. then shall the straight angle contained by AB, AC be equal to that contained by DE, DF. Because 'the angle contained by AB, AC is a straight angle, therefore BA, AC are in the same straight line BAC. Def. 7. For a like reason DE, DF are in the same straight line EDF. Then the straight line BAC can be made to fall on the straight line EDF, with the point A on the point D, and either with B on the same side of D as E, and C on the same side as F, or with C on the same side of D as E, and B on the same side as F, then in either case the straight angle contained by AB, AC coincides with that contained by DE, DF, and is equal to it, since magnitudes that can be made to coincide are equal. Q.E.D. COR. 1. The two straight angles which have the same arms, AB, AC, are equal. COR. 2. All right angles are equal to one another. For every right angle is half a straight angle, Def. 10. and the halves of equals are equal. Ax. h. Ax. I. Cancel Matter Inserted.] COR. 3. At a given point in a given straight line only one perpendicular can be drawn to that line. COR. 4. The complements of equal angles are equal. THEOR. 2. If a straight line stands upon another straight line, it makes the adjacent angles together equal to two right angles. Let the straight line AB stand upon the straight line CD: then shall the angles ABC, ABD be together equal to two right angles. Because the sum of the adjacent angles ABC, ABD is the angle contained by BC, BD, Def. 8. and this angle is a straight angle, since CBD is a straight line, Def. 7. therefore the angles ABC, ABD are together equal to a straight angle, that is, to two right angles. Def. 1o. Q.E.D. COR. All the angles made by any number of straight lines drawn from a point, each with the next following in order, are together equal to four right angles. Ex. 1. If two straight lines intersect, and one of the angles I. so formed is a right angle, show that the other three angles are also right angles. Cancel Matter Inserted.] * Ex. 2. The bisectors of the adjacent angles which one straight line makes with another include a right angle. THEOR. 3. If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line. Let the adjacent angles ABC, ABD which the straight line AB makes with the other two straight lines BC, BD be together equal to two right angles : then shall BC, BD be in one straight line. Because the sum of the adjacent angles ABC, ABD is the angle contained by BC, BD, Def. 8. and the sum of the angles ABC, ABD is two right angles, Hyp. therefore the angle contained by BC, BD is equal to two right angles, that is, to a straight angle, Def. 10. therefore BC, BD are in one straight line. Q.E.D Ax. C. * Exercises marked with an asterisk are worth remembering as results, or with a view to the solution of subsequent exercises. therefore the angles ABC, ABD are together equal to the angles EBC, EBD; Ax. c. therefore the angles ABC, ABD are together equal to two right angles. Q.E.D. COR. All the angles made by any number of straight lines drawn from a point, each with the next following in order, are together equal to four right angles. Ex. 1. If two straight lines intersect, and one of the four angles so formed is a right angle, shew that the other three angles are also right angles. * Ex. 2. The bisectors of the adjacent angles which one straight line makes with another include a right angle. THEOR. 3. If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line. Let the adjacent angles ABC, ABD which the straight line AB makes with the two other straight lines BC, BD be together equal to two right angles : then shall BC and BD be in one straight line. * Exercises marked with an asterisk are worth remembering as results, or with a view to the solution of subsequent exercises. |