equal (I. 3.), and the angle DCE is equal to DBA. But the angle BCF is evidently greater than DCE, it is consequently greater than DBA or ABC. In like manner, it may be shown, that if BC be produced, the exterior angle ACG is greater than CAB. But ACG is equal to its vertical angle BCF (Def. 10.), and hence BCF must be greater than either the angle CBA or CAB, PROP. XI. THEOR. Any two angles of a triangle are together less than two right angles. The two angles BAC and BCA of the triangle ABC are together less than two right angles. For produce the common side AC. And by the last proposition the exterior angle BCD is greater than CAB; add BCA to each, and the two angles BCD and BCA are greater than B A C D CAB and BCA, or CAB and BCA are together less than BCD and BCA, that is, less than two right angles (Def. 4). PROP. XII. THEOR. Every triangle has two acute angles. Let the triangle ABC have first a right angle at C. Then, by the last proposition, the angles ACB and CAB are less than two right an gles, and so are the angles ACB and ABC. Consequently the angles CAB and CBA A B C are each of them less than one right angle, or they are both acute. Next let the triangle have an obtuse angle ACB. The angles ACB and CAB, being less than two right an than one right angle. Consequently the angles CAB and CBA are both of them acute. Lastly, let the triangle have the angle at C acute. If one of the remaining angles, such as BAC, B be likewise acute, the two angles ACB and BAC are both of them acute. But if the C angle BAC be either obtuse or a right angle, it comes under the two former cases, and the other angles ABC and ACB are, therefore, acute. PROP. XIII. THEOR. If from a point without a straight line, two other straight lines be drawn to meet it; the nearer one will form on the same side a greater angle than that which is more remote. If straight lines CD, CE be drawn from the point C to the straight line AB; the angle ADC is greater than AEC. For ADC is the exterior angle of the triangle DCE, and is consequently (I. 10.) greater than the opposite interior angle CED. DE C B If the line CD be, therefore, supposed to turn about the point C in the direction of AB, the angle which it makes with the intercepted part of the line from A will continually diminish. Cor. Hence from any point only one straight line can be drawn, making a given angle on the same side with a given straight line; and hence also no more than one perpendicular can be let fall from a given point upon a given straight line. PROP. XIV. THEOR. In a triangle, that angle is the greater which lies opposite to a greater side. If a side BC of the triangle ABC be greater than BA; the opposite angle CAB is greater than BCA. B D C For make BD equal to BA, and join AD. The angle CAB is greater than DAB; but since BA is equal to BD, the angle DAB (I. 8.) is equal to ADB, and consequently CAB is greater than ADB. Again, the angle ADB, being an exterior angle of the triangle CAD, is (I. 10.) greater than ACD or ACB; wherefore the angle CAB is much greater than ACB. PROP. XV. THEOR. That side of a triangle is the greater which subtends a greater angle. If in the triangle ABC, the angle CAB be greater than ACB; its opposite side BC is greater than AB. For if BC be not greater than AB, it must be either equal or less. But it cannot be equal, be B cause the angle CAB would then be equal to ACB (I. 8); nor can BC be less than AB, for then AB would be greater than BC, and consequently (I. 14.) the angle C ACB would be greater than CAB, or CAB less than ACB, which is still more absurd. The side BC being thus neither equal to AB, nor less than it, must therefore be greater than AB. PROP. XVI. THEOR. Two sides of a triangle are together greater than the third side. The two sides AB and BC of the triangle ABC are together greater than the third side AC. For produce AB until DB be equal to the side BC, and join CD. 1 D side AD is greater than AC (I. 15.); and since AD is equal to AB and BD or to AB and BC, the two sides AB and BC are together greater than the third AC. PROP. XVII. THEOR. The difference between two sides of a triangle is less than the third side. Let the side AC be greater than AB, and from it cut off a part AE equal to AB; the remain der EC is less than the third side BC. B there remains BC greater than EC, or EC is less than AB. PROP. XVIII. THEOR. The shortest line that can be drawn between two given points, is a straight line. Let the points A and B be connected by straight lines joining an intermediate point C; and the two sides AC and BC of the triangle ACB are greater than AB (I. 16.) Now let a third point D be interposed between A and C; and because AD and DC are together greater than AC, add BC to both, and the three lines AD, DC, and CB are greater than AC and BC, and consequently much greater than AB. Again suppose a fourth point E to connect B with C; and the sides BE and CE of the triangle BCE being greater H G C D than BC, the four straight lines AD, DC, CE, and EB are toge E ther trebly greater than AB. By B thusrepeatedly multiplying the in- A terjacent points, two sides of a triangle will at each successive step come in place of a third side, and consequently the aggregate polygonal or crooked line AFDGCHEIB will acquire continually some farther extension. Nay, since there is no limit to the possible number of those connecting points, they may approach each other nearer than any assignable interval; and consequently the proposition is also true in that extreme case where the boundary is a curve line, or of which no portion can be deemed rectilineal. PROP. XIX. THEOR. Two straight lines drawn to a point within a triangle from the extremities of its base, are together less than the sides of the triangle, but contain a greater angle. |