AC will fall along DF, since the angle BAC is equal to the angle EDF, Hyp. and, AC falling along DF, C will fall on F, since AC is equal to DF. Hyp. Hence, B falling on E, and C on F, BC will coincide with Ax. 2. EF, and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it, Ax. I. the side BC equal to the side EF, the angle ACB to the angle DFF, and the angle ABC to the angle DEF. Q.E.D. Ex. 4. The straight line which bisects the vertical angle of an isosceles triangle bisects the base. Ex. 5. Any point on the bisector of the vertical angle of an isosceles triangle is equidistant from the extremities of the base. Ex. 6. The straight line which bisects the vertical angle of an isosceles triangle is perpendicular to the base. Ex. 7. Any point D is taken on the bisector of an angle BAC; prove that, if AB is equal to AC, then the angle ADB is equal to the angle ADC. Ex. 8. The straight lines drawn from the extremities of the base of an isosceles triangle to the middle points of the opposite sides are equal to one another. Ex. 9. On one arm of an angle whose vertex is A points B and D are taken, and on the other arm points C and E, such that AB is equal to AC, and AD to AE: shew that BE is equal to CD. THEOR. 6. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides between the vertices of these angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to the equal angles. Let ABC, DEF be two triangles having the angle ABC equal to the angle DEF, the angle ACB to the angle DFE, and the side BC to the side EF: E then shall the triangles be identically equal, having the angle BAC equal to the angle EDF, the side AC to the side DF, and the side AB to the side DE. Let the triangle ABC be applied to the triangle DEF, so that the point B may fall on the point E, the side BC along the side FF, and the point A on the same side of EF as the point D; then C will fall on F, since BC is equal to EF, Hyp. BA will fall along ED, since the angle CBA is equal to the angle FED, Hyp. and CA will fall along FD, since the angle BCA is equal to the angle EFD; Hyp. hence A, which is the point of intersection of BA and CA, will fall on D, which is the point of intersection of ED and FD, and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it, Ax. I. the angle BAC equal to the angle EDF, the side AC to the side. DF, and the side AB to the side DE. Q.E.D. Ex. 10. If the bisector of an angle of a triangle is also perpendicular to the opposite side, the triangle is isosceles. THEOR. 7. If two sides of a triangle are equal, the angles opposite to those sides are equal. Let ABC be a triangle having the side AB equal to the side AC: then shall the angle ACB be equal to the angle ABC. Let A'B'C' be a triangle identically equal to the triangle ABC, the points A',B',C' corresponding respectively to the points A,B,C. Then in the triangles ABC, A'C'B', AB is equal to A'C', since it is equal to AC, Hyp. and AC is equal to A'B', since it is equal to AB, Нур. and the angle BAC is equal to the angle C'A'B', therefore the angle ACB, which is opposite to the side AB, is equal to the angle A'B'C', which is opposite to the side A'C', I. 5. that is, the angle ACB is equal to the angle ABC. COR. If a triangle is equilateral, it is also equiangular. Q.E.D. Ex. 11. Prove Theor. 7 by comparing the triangles into which the bisector of the vertical angle divides the isosceles triangle. THEOR. 8. If two angles of a triangle are equal, the sides opposite to those angles are equal. Let ABC be a triangle having the angle ABC equal to the angle ACB: A A then shall the side AC be equal to the side AB. Let A'B'C' be a triangle identically equal to the triangle ABC, the points A',B',C' corresponding respectively to the points A,B,C. Then in the triangles ABC, A'C'B', the angle ABC is equal to the angle A'C'B', since it is equal to the angle ACB, Hyp. and the angle ACB is equal to the angle A'B'C', since it is equal to the angle ABC, and BC is equal to C'B', Hyp. therefore the side AC, which is opposite to the angle ABC, is equal to the side A'B', which is opposite to the angle A'C'B', I. 6. that is, the side AC is equal to the side AB. COR. If a triangle is equiangular, it is also equilateral. Q.E.D. Ex. 12. If the angles at the base of an isosceles triangle are bisected, the bisectors and the base form an isosceles triangle. THEOR. 9. If any side of a triangle is produced, the exterior angle is greater than either of the interior opposite angles. Let ABC be a triangle having the side BC produced to D: E 4 F then shall the exterior angle ACD be greater than either of the interior opposite angles BAC, ABC. Let E be the middle point of AC; join BE, and produce BE to F, making EF equal to BE, join FC. |