In any triangle ABC, one of its legs, as BC, being produced towards D, it will make the external angle A CD equal to the two internal opposite angles taken together; viz. to B and A. Through C, let CE be drawn parallel to AB; then since BD cuts the two parallel lines BA, CE, the angle ECD=B (by part 3 of the last theo.); and again, since AC cuts the same parallels, the angle ACE=A (by part 2 of the last), therefore ECD+ACE=A CD= B+A. Q. E. D. Cor. 1. Hence, if a triangle have its exterior angle and one of its opposite interior angles double of those in another triangle, its remaining opposite interior angle will also be double of the corresponding angle in the other.t That invaluable instrument, Hadley's Quadrant, is founded on this corollary, annexed as an obvious consequence of the theorem. A ray of light SA (Pl. 14. fig. 2) from the sun, against the mirror at A, is reflected at an angle equal to its incidence; and now striking the half-silvered glass at C, it is again reflected to E, where the eye likewise receives, through the transparent part of that glass, a direct ray from the boundary of the horizon. -- . Hence, the triangle AEC has its exterior angle ECD and one of its interior angles CAE respectively double of the exterior angle BCD and the interior angle CAB of the triangle * For an excellent demonstration of this theorem (by the motion of the straight line crossing the parallel lines about a point in one of them), the reader will consult Leslie's Geometry, Prop. 23, page 26, . . . t This corollary, with the following demonstration, is found in Leslie's Geometry, pages 32 and 406. * : . . . . .”. ABC; wherefore the remaining interior angle AEC, or SEZ, is double of ABC; that is, the altitude of the sun above the horizon is double of the inclination of the two mirrors. But the glass at C remaining fixed, the mirror at A is attached to a moveable index, which marks their inclination. The same instrument, in its most improved state, and fitted with a telescope, forms the sextant, which, being admirably calculated for measuring angles in general, has rendered the most important services to geography and navigation. THEOREM. W. PL. 1. fig. 23. In any triangle ABC, all the three angles, taken together, are equal to two right angles, viz. A+B+ACB= two right angles. Produce CB to any distance, as D, then (by the last) A CD = B+A ; to both add ACB ; then A CD+ACB=A+B+ ACB; but A CD+ACB= two right angles (by theo. 1); therefore the three angles A+B+ACB= two right angles. Q.E.D. Cor. 1. Hence if one angle of a triangle be known, the sum of the other two is also known; for since the three angles of every triangle contain two right ones, or 180 degrees, therefore 180— the given angle will be equal to the sum of the other two; or 180—the sum of two given angles gives the other one. Cor. 2. In every right-angled triangle, the two acute angles are = 90 degrees, or to one right angle; therefore 90— one acute angle gives the other. If in any two triangles, ABC, DEF, there be two sides AB, AC in the one severally equal to DE, DF in the other, and the angle A contained between the two sides in the one equal to D in the other; then the remaining angles of the one will be severally equal to those of the other, viz. B=E, o: C=F; and the base of the one BC will be equal to EF, that of the Other. If the triangle ABC be supposed to be laid on the triangle DEF, so as to make the points A and B coincide with D and E, which they will do, because AB-DE (by the hypothesis); and since the angle A=D, the line AC will fall along DF, and inasmuch as they are supposed equal, C will fall in F; seeing therefore the three points of one coincide with those of the other triangle, they are manifestly equal to each other; therefore the angle B=E, and C=F, and BC= EF, Q. E. D. LEMMA. * If two sides of a triangle abc be equal to each other, that is, ac=cb, the angles which are opposite to those equal sides will also be equal to each other ; viz. a+b. For let the triangle abc be divided into two triangles acd, dcb, by making the angle acd=dcb (by postulate 4); then because ac=bc, and cd common (by the last), the triangle adc-dcb ; and therefore the angle a=b. Q. E. D. . Cor. Hence if from any point in a perpendicular which bisects a given line there be drawn right lines to the extremities of the given one, they with it will form an isosceles triangle. THEOREM WII. The angle BCD at the centre of a circle A BED is double the angle BAD at the circumference, standing upon the same arc BED. Through the point A, and the centre C, draw the line ACE; then the angle ECD=CAD+CDA (by theo. 4); but since AC=CD, being radii of the same circle, it is plain (by the preceding lemma) that the angles subtended by them will be also equal, and that their sum is double to either of them, that is, DAC+ADC is double to CAD, and therefore ECD is double to CAD ; after the same manner BCE is double to CAB, wherefore BCE-HECD, or BCD, is double to BAC + CAD, or to BAD. Q. E. D. Cor. 1. Hence an angle at the circumference is measured by half the are it subtends or stands on. Fig. 26. Cor. 2. Hence all angles at the circumference of a circle which stand on the same chord as AB are equal to each other, for they are all measured by half the arc they stand on, viz. by half the arc AB Fig. 26. Cor. 3. Hence an angle in a segment greater than a semicircle is less than a right angle; thus ADB is measured by half the arc AB; but as the arc AB is less than a semicircle, therefore half the arc AB, or the angle ADB, is less than half a semicircle, and consequently less than a right angle. Fig. 27. Cor. 4. An angle in a segment less than a semicircle is greater than a right angle; for since the arc AEC is greater than a semicircle, its half, which is the measure of the angle ABC, C must be greater than half a semicircle, that is, greater than right angle. Fig. 28. Cor. 5. An angle in a semicircle is a right angle, for the measure of the angle ABD is half of a semicircle AED, and therefore a right angle. THEOREM VIII. If from the centre C of a circle ABE there be let fall the perpendicular CD on the chord AB, it will bisect it in the point D. Let the lines AC and CB be drawn from the centre to the extremities of the chord; then since CA=CB, the angles CAB =CBA (by the lemma). But the triangles ADC, BDC are right-angled ones, since the line CD is a perpendicular; and so the angle ACD=DCB (by cor. 2, theo. 5); then have we AC, CD, and the angle A CD in one triangle severally equal to CB, CD, and the angle BCD in the other; therefore (by theo. 6) AD=DB. Q. E. D. * - * Cor. Hence it follows, that any line bisecting a chord at right angles is a diameter; for a line drawn from the centre perpendicular to a chord bisects that chord at right angles; therefore, conversely, a line bisecting a chord at right angles must pass through the centre, and consequently be a diameter. THEOREM IX. If from the centre of a circle ABE there be drawn a perpendicular CD on the chord AB, and produced till it meets the circle in F, that line CF will bisect the arc AB in the point F. Let the lines AF and BF be drawn; then in the triangles ADF, BDF, AD= BD (by the last); DF is common, and the angle ADF= BDF, being both right, for CD or DF is a perpendicular. Therefore (by theo. 6) AP= FB; but in the same circle, equal lines are chords of equal arcs, since they measure them (by def. 19); whence the arc AF =FB, and so AFB is bisected in F by the line CF. Cor. Hence the sine of an arc is half the chord of twice that arc. For AD is the sine of the arc AF (by des. 20), AF is half the arc, and AD half the chord AB (by theo. 8); therefore the corollary is plain. THEOREM X. PL. 1. fig. 30. In any triangle ABD, the half of each side is the sine of the opposite angle. Let the circle ADB be drawn through the points A, B, D; then the angle DAB is measured by half the arc BKD (by cor. 1, theo. 7), viz. the arc BK is the measure of the angle BAD; therefore (by cor. to the last) BE, the half of BD, is the sine of BAD: in the same way may be proved that half of AD is the sine of ABD, and the half of AB the sine of ADB. Q. E. D. THEOREM XI. PL. 1. fig. 22. If a right line GH cut two other right lines AB, CD, so as to make the alternate angles AEF, EFD equal to each other, then the lines AB and CD will be parallel. If it be denied that AB is parallel to CD, let IK be parallel to it; then IEF=(EFD)=AEF(by part 2, theo. 3), a greater to a less, which is absurd, whence IK is not parallel; and the like we can prove of all other lines but AB; therefore AB is parallel to CD. Q. E. D. THEOREM XII. If two equal and parallel lines AB, CD, be joined by two other lines AD, BC, those shall be also equal and parallel. Let the diameter or diagonal BD be drawn, and we will have the triangles ABD, CBD, whereof AB in one is a to CD in the other, BD common to both, and the angle ABD=CDB (by part 2, theo. 3); therefore (by theo. 6) AD=CB, and the angle CBD=ADB ; and thence the lines AD and BC are parallel, by the preceding theorem. Cor. 1. Hence the quadrilateral figure ABCD is a parallelogram, and the diagonal BD bisects the same, inasmuch as the triangle ABD=BCD, as now proved. Cor. 2. Hence also the triangle ABD on the same base AB, and between the same parallels with the parallelogram ABCD, is half the parallelogram. Cor. 3. It is hence also plain that the opposite sides of a parallelogram are equal; for it has been proved that, ABCD being a parallelogram, AB wo = CD, and AD=BC. 2 |