Next; let the angle ABC be equal to the angle DEF, and (case II.) the side AB equal to the side DE; D B E F Construction. If BC is not equal to EF, one of them must be greater than the other; suppose BC to be the greater, and from it cut off BG equal to EF: (prop. 3). Join AG. Demonstration. In the triangles ABG, DEF, AB is equal to DE, and BG to EF, and the angle ABG to the angle DEF, therefore the triangles are equal in all respects; (prop. 4) and therefore the angle AGB is equal to the angle DFE : but the angle ACB is equal to the angle DFE; (hyp.) therefore the angle AGB is equal to the angle ACB; that is, an exterior angle equal to an interior opposite angle, which is impossible. (prop. 16). Hence BC is not unequal to EF, i.e., it is equal to it. AB is equal to DE, BC to EF, and the angle ABC to the therefore the triangles ABC, DEF are equal in all respects. Exercises. (prop. 4). Q.E.D. 1. The perpendiculars let fall on the sides of a triangle from any point in the line bisecting the angle contained by them are equal. 2. Find a point in a given straight line such that the perpendiculars let fall from it upon two given straight lines are equal. 3. Draw through one of three given points a straight line such that the perpendiculars let fall upon it from the other points shall be equal and on opposite sides. 4. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to one pair of equal sides right angles; then shall the triangles be equal in all respects. Miscellaneous Exercises. 1. On a given straight line describe an isosceles triangle each of whose sides shall be double of the given line: also another triangle, each of whose sides shall be four times the given line. 2. Two straight lines are drawn bisecting each other at right angles; shew that the figure formed by joining their extremities has all its sides equal. 3. A straight line DEF is drawn through the middle point D, of the side BC of the triangle ABC at right angles to it, and it cuts the sides BA, CA or those sides produced in E and F; shew that EB and EC are equal to each other, and also FB and FC. 4. The angles at the base of an isosceles triangle are bisected by two straight lines which intersect in a point D; shew that the vertical angle is bisected by the straight line joining it to the point D. 5. Shew that the diameter is the longest line in a circle. 6. ABC is an isosceles triangle in which each of the sides is greater than the base; if D be any point in AB, shew that AB is greater than CD. 7. The straight lines which join the extremities of any two diameters of a circle are equal to each other. 8. Two straight lines meet at A; draw through any point B outside the lines a straight line which shall cut them in C and D, so that ACD shall be an isosceles triangle. 9. In an isosceles triangle, a straight line drawn from the vertex to any point in the base is less than the side of the triangle. 10. The diagonals of a parallelogram are together greater than half the sum of the sides. 11. The diagonals of a square are equal to each other, bisect each other and are at right angles to each other. 12. If two circles intersect at A and B, prove that the straight line AB is bisected by the straight line joining the centres of the circles. 13. A straight line is drawn in a circle terminated by its circumference; shew (1) that the straight line drawn from the centre of the circle to the middle point of the line is at right angles to it; and (2) that a straight line drawn from the centre at right angles to the straight line will bisect it. 14. The portions of a straight line intercepted between two concentric circles are equal to one another. 15. If a side AC of any triangle ABC be bisected in E, and BE be joined and produced to F so that EF is equal to BE; and if AF and FC be joined ; shew that the figure AFCB is double of the triangle ABC. |