TRIANGLES THEOREM 10 The exterior angle, made by producing one side of a triangle, equals the sum of the two interior and opposite angles; and the three interior angles are together equal to two right angles. Hypothesis.—ABC is a ▲ having BC produced to D. To prove that (1) LACD = L A+ LB. (2) LA+B+ ACB = two rt. Ls. Construction.-Through C draw CE || AB. Cor. The exterior angle of a triangle is greater than either of the interior and opposite angles. 56.-Exercises 1. Prove Theorem 10 by means of a st. line drawn through the vertex || the base. 2. If two As have two 4s of one respectively equal to two ≤s of the other, the third of one is equal to the third of the other. 3. The sum of the 4s of a quadrilateral is equal to four rt. s. 4. The sum of the s of a pentagon is six rt. ≤s. 5. Each of a equilateral ▲ is an of 60°. 6. Find a point B in a given st. line CD such that, if AB be drawn to B from a given point A, the ABC will equal a given 2. 7. Show that the bisectors of the two acute 4s of a rt.-2d A contain an of 135°. 8. If both pairs of opposites of a quadrilateral are equal, the quadrilateral is a ||gm. 9. C is the middle point of the st. line AB. CD is drawn in any direction and equal to CA or CB. Prove that ADB is a rt. 4. 10. On AB, AC, sides of a ▲ ABC, equilateral As ABD, ACE are described externally. Show that DC = BE. 11. AB is any chord of a O. AB is produced to C so circle of which the centre is = that BC BO. CO is joined, cutting the circle at D and is produced to cut it again at E. Show that AOE = three times / BCD. 12. If the exterior s at B and C of a AABC be bisected and the bisectors be produced to meet at D, the / BDC equals half the sum of 4s ABC, ACB. 13. Show that a ▲ must have at least two acute <s. 14. In an acute-zd▲ show that the from a vertex to the opposite side cannot fall outside of the A. 15. In an obtuse-/d show that the L from the vertex of the obtuse on the opposite side falls within the A, but that the from the vertex of either acute on the opposite side falls outside of the A. 16. In a rt.-≤d ▲ where do the 1s from the vertices on the opposite sides fall? 17. Only one can be drawn from a given point to a given st. line. 18. Not more than two st. lines each equal to the same given st. line can be drawn from a given point to a given st. line. 19. D is a point taken within the ABC. Join DB, DC; and show, by producing BD to meet AC, that < BDC > < BAC. 20. With compasses and ruler only, construct the following Ls: -30°, 15°, 120°, 105°, 75°, 67, 150°, 195°, 210°, 240°, 255°, 285°, 30°, - 75', - 135°. - 21. If a transversal cut two st. lines so as to make the interior s on one side of the transversal together less than two rt. Ls, the two lines when produced shall meet on that side of the transversal. 22. The bisector of the exterior vertical ▲ of an isosceles A is to the base. 23. Give a proof for the following method of drawing a line through P LAB: First place the set-square in the position shown by the dotted line, with its hypotenuse along AB. Place a ruler along one of the sides of the set-square and hold it firmly in that position. Rotate the set-square through its right 4, thus bringing the other side against the ruler, and slide the set-square along the ruler to the position shown by the shaded ▲. A line drawn through P, along the hypotenuse of the set-square, is perpendicular to AB. THEOREM 11 If one side of a triangle is greater than another side, the angle opposite the greater side is greater than the angle opposite the less side. B Hypothesis. ABC is a ▲ having AB > AC. Construction.-From AB cut off AD = AC. Join DC. BD is produced to A, .. exterior ADC > interior and opposite Z DBC. (I-10, Cor., p. 45.) But ACB > < ADC; much more .. is < ACB > < ABC. |