THEOREM 12 (Converse of Theorem 11) If one angle of a triangle is greater than another angle of the same triangle, the side opposite the greater angle is greater than the side opposite the less. But this also is not so, . AC is not <AB. 1. The perpendicular is the shortest st. line that can be drawn from a given point to a given straight line. The length of the from a given point to a given st. line is called the distance of the point from the line. 2. ABCD is a quadrilateral, of which AD is the longest side, and BC the shortest. Show that B > < D, and that <C> <A. 3. The hypotenuse of a rt.-d ▲ is greater than either of the other two sides. 4. A st. line drawn from the vertex of an isosceles ▲ to any point in the base is less than. either of the equal sides. x 5. A st. line drawn from the vertex of an isosceles ▲ to any point in the base produced is greater than either of the equal sides. x 6. If one side of a ▲ be less than another, opposite the less side is acute. the 7. If D be any point in the side BC of a ▲ ABC, the greater of the sides AB, AC, is greater than AD. 8. AB is drawn from AL CD. E, F are two points in CD on the same side of B, and such that BE BF. Show that AEAF. Prove the same proposition when E, F are on opposite sides of B. * 9. ABC is a ▲ having AB > AC. The Show that BD > DC. bisector of A Give a general If the bisectors of meets BC at D. 10. ABC is a A having AB > AC. 11. Prove Theorem 11 from the following construction: Bisect / A by AD which meets BC at D; from AB cut off AE = AC, and join ED. E 12. The s at the ends of the greatest side of a ▲ are acute. 13. If AB > AD in the gm ABCD, ADB > < BDC. THEOREM 13 (Converse of Theorem 3) If two angles of a triangle are equal to each other, the sides opposite these equal angles are equal to each other. 1. An equiangular ▲ is equilateral. 2. BD, CD bisect the s ABC, ACB at the base of an isosceles ABC. Show that ▲ DBC is isosceles. 3. ABC is a ▲ having AB, AC produced to D, E respectively. The exterior LS DBC, ECB are bisected by = BF, CF, which meet at F. Show that, if FB FC, the 4. On the same side of AB the two As ACB, ADB have AC = BD, AD = = BE. AE 5. On a given base construct a ▲ having one of the s at the base equal to a given ▲, and the sum of the sides equal to a given st. line. 6. On a given base construct a ▲ having one of the s at the base equal to a given and the difference of the sides equal to a given st. line. 7. If the bisector of an exterior of a Abe to the opposite side, the A is isosceles. 8. Through a point on the bisector of an ட் a line is drawn to one of the arms. Prove that the A thus formed is isosceles. 9. A st. line drawn to BC, the base of an isosceles A ABC, cuts AB at X and CA produced at Y. Show that AXY is an isosceles A. at C. Through 10. ACB is a rt.- d▲ having the rt. X, the middle point of AC, XY is drawn || at Y. Show that Y is the middle point of AB. 11. The middle point of the hypotenuse of a rt.-≤d ▲ is equidistant from the three vertices. 12. The st. line joining the middle points of two sides of a A is to the third side. 13. Construct a rt.- ≤ d ▲, having the hypotenuse equal to one given st. line, and the sum of the other two sides equal to another given st. line. . 14. If one of a A equals the sum of the other two, show that the ▲ is a rt.- ≤ d A. N THIRD CASE OF THE CONGRUENCE OF TRIANGLES THEOREM 14 If two triangles have two angles and a side of one respectively equal to two angles and the corresponding side of the other, the triangles are congruent. Hypothesis. —ABC, DEF are two As having A = LD, LB LE, and BC = EF. To prove that ▲ ABC=▲ DEF and B = LE, :: <A+ LB = < D + 2 E. But A+B+/C=<D+<E+ ≤ F. (I—10, p. 45.) Apply = < F. ABC to ▲ DEF so that BC coincides with the equal side EF. .. BA falls along ED, and A is on the line ED. .. CA falls along FD, and A is on the line FD. But D is the only point common to ED and FD, . A falls on D. :. ▲ ABC coincides with ▲ DEF, and. A ABC A DEF. |