255. Theorem. If two parallel lines are cut by a transversal, the exterior interior angles are equal. Prove by the indirect method, using the figure of § 254. 256. Theorem. If two parallel lines are cut by a transversal, the consecutive interior angles are supplementary. Prove by the indirect method. 257. Theorem. If two parallel lines are cut by a transversal, the alternate exterior angles are equal. Prove by the indirect method. 258. Theorem. If two triangles have two sides of one equal respectively to two sides of the other but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second Given triangles DEF and ABC having DE=AB, EF=BC, and <E><B. To prove that DF>AC. Proof. Place AABC upon ADEF so that AB coincides with DE and AABC takes the position of ADEK. Bisect ZKEF by the line EH and draw KH. 259. Theorem. If two triangles have two sides of one equal to two sides of the other but the third side of the first greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second. Given the triangles DEF and ABC having DE=AB, EF BC and DF>AC. = To prove <E><B. Proof. It must be true that ZE=ZB or ZE<ZB or <E><B. Suppose ZE ZB. Then DF = AC. Suppose ZE<<B. Then DF<AC. Why? § 258 Since it was given that DF>AC, neither of these suppositions can be true. :: <E><B. 260. Proof by Analysis. When the proof of an exercise is not readily seen, the figure should be analyzed to find conditions under which the conclusion must be true. This may be illustrated by the following exercise. EXERCISE If the middle point of one side of a triangle is equidistant from the three vertices, the triangle is a right triangle. Given that B is equidistant from A, C, and D. To prove that AACD is a right triangle. Analysis and proof. AACD is a right triangle if ZD is a right angle. ZD is a right angle if ZD=ZA+ZC. ZD=ZA+2C if Zz+Zx = ZA+ZC. 2x+2x=ZA+ZC if Zz=ZA and x = ZC. Zz=ZA and Zx = ≤C if ▲ ABD and CBD are isosceles. A AABD and CBD are isosceles if AB=BD and BC = BD. B But AB=BD and BC=BD since it is given that B is equidistant from A, C, and D. Therefore all the conditions are true and AACD is a right triangle. 261. Proof by Synthesis. By the synthetic method of proof, a truth is proved by putting together other truths which have previously been proved or established. This is the method most commonly used. Nearly all of the theorems and exercises in Chapter I are proved by this method. In many exercises the proof is almost obvious from an examination of the figure and the hypothesis. In more difficult exercises, it is well to resort to analysis in order to discover the proof. QUESTIONS 1. If one of the acute angles of a right triangle is given, how can the other be found? 2. If one angle of a triangle is a right angle, what relation exists between the other two angles? 3. How many altitudes has a triangle? How many medians? 4. How many right angles can there be in a triangle? How many obtuse angles? How many acute angles? 5. How many degrees in each angle of an equilateral triangle? 6. A right triangle is isosceles. How many degrees in each angle? 7. A right triangle has acute angles of 30° and 60°. What is the relation between the hypotenuse and the shorter side? 8. What does the altitude from the vertex of an isosceles triangle do to the base? To the vertex angle? To the triangle? 9. State several facts about the altitudes of an equilateral triangle. 10. Where do the three medians of any triangle meet? 11. In what quadrilaterals are the diagonals perpendicular to each other? 12. In what quadrilaterals do the diagonals bisect each other? In what ones do they bisect the angles? 13. State as many facts as you can about the diagonals of a square. Of a rhombus. Of a rectangle. 14. State different methods of proving two triangles equal. 15. State methods of proving two right triangles equal. 16. Give several conditions which show that two lines are equal. 17. Give several conditions which show that two angles are equal. 18. What are some of the methods of proving that two lines are parallel? EXERCISES INVOLVING EQUAL LINES 1. The sum of the perpendiculars from any point in the base of an isosceles triangle to the two equal sides equals the perpendicular from either end of the base to the opposite side. B F 2. If from any point in an equileteral triangle perpendiculars are drawn to the three sides, the sum of these perpendiculars is constant, and equal to the altitude of the triangle. 3. If upon the three sides of any triangle, equilateral triangles are constructed outward, and a line is drawn from each vertex of the triangle to the farthest vertex of the equilateral triangle on the opposite side, these three lines are equal. Prove that AD=FC=BE. 4. The difference of the perpendiculars from any point in the base produced of an isosceles triangle to the equal sides equals the perpendicular from either end of the base to the opposite side. 5. Triangle ABC is isosceles. AD=CF. Prove that DF is bisected by the base AC. EXERCISES INVOLVING EQUAL ANGLES E B D A E F 1. Lines BD and CD bisect the base angles of the isosceles triangle BEC. Prove that ZABE = ZBDC. 2. In the triangle ABC, DB 1 AC. Prove that ≤x-2y= ZA — ZC. 3. Figure AHK is a right triangle with ZH the right angle. HD is an altitude and HC a median. Prove 4. The angle formed by the bisectors of two exterior D angles of a triangle equals half the sum of the interior angles at the same vertices. B E 5. In a right triangle, prove that the angle formed by the bisector of the right angle and the median drawn to the hypotenuse equals half the difference of the two acute angles. 6. In the isosceles triangle ACE, FB and FD are perpendicular to the equal sides. Prove that x=z. 7. In a right triangle, the angle formed by the median and the altitude drawn to the hypotenuse is bisected by the bisector of the right angle. 8. In the figure, AC bisects ZBAE, and EC bisects BED. Prove that ≤C = 1⁄2 ZB. A B C E F D EXERCISES INVOLVING PARALLELS AND PERPENDICULARS 1. If one of the equal sides of an isosceles triangle is produced through the vertex by its own length, the line joining the end of the side produced to the nearest end of the base is perpendicular to the base. 2. If medians are drawn to the equal sides of an isosceles triangle, the line joining the points of meeting is parallel to the base. 3. If two opposite angles of a quadrilateral are right angles, the bisectors of the other two angles are parallel. A B 4. Prove that the following method of drawing a perpendicular through P to AB is correct. First, place the triangle in the position shown by the dotted lines, with its hypotenuse along AB. Second, place a straightedge along the side of the triangle, as shown, and hold it in that position. Third, place the triangle in the position shown by the shaded tri angle. Fourth, draw a line through P along the hypotenuse of the triangle. This is the perpendicular required. A -B 5. Prove that the following method of drawing a parallel to AB through P is correct. First, place the triangle with the hypotenuse along the line AB. Second, place a straightedge along a side of the triangle, and hold in that position while the triangle is placed so that its hypotenuse passes through P. The line through P along the hypotenuse is parallel to AB. |