Proof (of b). angle A G H is equal to angle GHD (h). angle AGH is equal to angle E G B. (i) angle E G B is equal to angle G H D. Proof (of c). angle E GB is equal to angle GHD (i). add the angle B G H to each of these. Then, by Axiom 2a, Q. E. D. (k) angles EG B, B G H are equal to angles BGH, GHD. But, by Euc. I. 13, EGB, BGH are together equal to two right angles. ... angles BGH, G H D are together equal to two right angles. Q. E. D. THEOREM (Euclid I. 30). Repeat. The enunciations of Euc. I. 27, I. 29, and Axiom 1. General Enunciation. Straight lines which are parallel to the same straight line are parallel to one another. Required. To prove that line AB is parallel to line CD. Construction. (c) Let the straight line GHK fall on the lines AB, CD, and EF by Euc. I. 29, (d) angle A G H is equal to angle G H F. Again, EF is parallel to CD (b), and GHK falls upon them (c). .. by Euc. I. 29, (e) angle GHF is equal to angle HK D. .. by conclusion (d) and (e), and Axiom 1, angle A G H is equal to angle HKD. But angles A G H and HKD are alternate angles. .. by Euc. I. 27, AB is parallel to CD. Q. E. D. PROBLEM (Euclid I. 31). General Enunciation. To draw a straight line through a given point parallel to a given straight line. Particular Enunciation. Given. The line AB, and the point C. Required. To draw a line through C, parallel A Construction. In AB take any point D. C B E Fig 2. A (a) At the point Cin the straight line CD, (by Euc. I. 23), make the angle DCE equal to the angle CDB (Fig. 2). Produce the straight line EC to any point F (Fig. 3). Proof. The straight line CD meets the two lines EF and A B, and makes the angle ECD equal to the alternate angle CDB (a). F A. Fig. 3 by Euc. I. 27, EF is parallel to A B. B Wherefore a straight line EF has been drawn through C, parallel to AB. Q. E. D. EXERCISE.-Given.-The triangle ABC. B Required. To draw a line which shall bisect BC (see Euc. I. 10, page 32), and be parallel to A C. THEOREM (Euclid I. 32, First Part). Repeat. The enunciations of Euc. I. 29. General Enunciation. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles. (Compare this with Euc. I. 16, page 56, which proves that the exterior angle is greater than either of the interior opposite angles.) Particular Enunciation. Given. The triangle A B C, having the side AC produced to D. A B Required. To prove that angle A CD is equal to the two interior opposite angles, CA B, ABC. A E B Construction. (a) Through the point C draw (by Euc. I. 31) CE parallel to A B. |