THEOREM XX. 67. The sum of the interior angles of a polygon is equal to twice as many right angles as it has sides minus two. Let ABCDEF be the given polygon; the sum of all the interior angles A, B, C, D, E, F, is equal to twice as many right angles as the figure has sides minus two. F C B E D For if from any vertex A, diagonals À C, AD, AE, are drawn, the polygon will be divided into as many triangles as it has sides minus two; and the sum of the angles of each triangle is equal to two right angles (33); therefore the sum of the angles of all the triangles, that is, the sum of the interior angles of the polygon, is equal to twice as many right angles as the polygon has sides minus two. PRACTICAL QUESTIONS. 1. Do two lines that do not meet form an angle with each other? Two lines not in the same plane? 2. Does the magnitude of an angle depend upon the length of its sides? 3. If a right angle is 90°, what is the complement of an angle of 27° ? of 51° of 91° of 153°? What is the supplement of an angle of 13° ? of 83° of 97° ? of 217° ? 4. If three of four angles formed at a point on the same side of a straight line, in the same plane, contain respectively 15°, 27°, and 99°, how many degrees does the fourth angle contain? 5. If five of six angles formed in a plane about a point are respectively 11o, 53°, 74°, 19°, and 117°, how many degrees are there in the sixth angle? 6. On opposite sides of a line AB are two lines making with A B, at the point A, the first an angle of 29°, and the second an angle of 61°; how are these two lines related? 7. Can two polygons, each not equilateral, be mutually equilateral? 8. Can two polygons, each not equiangular, be mutually equiangular? 9. If two angles of a triangle are respectively 32° and 43°, how many degrees are there in the remaining angle? 10. If one acute angle of a right triangle is 24°, how many degrees are there in the other acute angle? 11. How many degrees in each angle of an equiangular triangle? 12. How many degrees in each angle at the base of an isosceles triangle whose vertical angle is 14° ? 13. How many degrees in each acute angle of a right-angled isosceles triangle ? 14. If one of the angles at the base of an isosceles triangle is double the angle at the vertex, how many degrees in each? 15. If the angle at the vertex of an isosceles triangle is double one of the angles at base, how many degrees in each? 16. Two triangles mutually equilateral are equiangular (48). Are two triangles mutually equiangular also equilateral ? 17. Is a square a parallelogram? Is a parallelogram a square? 18. Is a rectangle a parallelogram? Is a parallelogram a rectangle? 19. How many sides equal to one another can there be in a trapezoid? How many in a trapezium? 20. How many degrees in each angle of an equiangular pentagon ? an equiangular hexagon? octagon? decagon? dodecagon ? 21. If the parallel sides of a trapezoid are respectively 8 feet and 13 feet in length, how long is the line joining the middle points of the other two sides? 22. If one of the angles of a parallelogram is 120°, how many degrees are there in each of the other angles? EXERCISES. The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the propositions in the following books, and can be omitted if thought best. 68. Two angles whose sides have, one pair the same, the other opposite directions, are supplements of each other. (12.) (8.) 69. Any side of a triangle is less than the sum, but greater than the difference, of the other two. (Axiom 9.) 70. The sum of the lines drawn from a point within a triangle to the extremities of one of the sides is less than the sum of the other two sides. Produce one of the lines to the side of the triangle. (Axiom 9.) 71. The angle included by the lines drawn from a point within a triangle to the extremities of one of the sides is greater than the angle included by the other two sides. Produce as in (70). (39.) 72. The angle at the base of an isosceles triangle being one fourth of the angle at the vertex, if a perpendicular is drawn to the base at its extreme point meeting the opposite side produced, the triangle formed by the perpendicular, the side produced, and the remaining side of the triangle is equilateral. 73. If an isosceles and an equilateral triangle have the same base, and if the vertex of the inner triangle is equally distant from the vertex of the outer and the extremities of the base, then the angle at the base of the isosceles triangle isor of its vertical angle, according as it is the inner or the outer triangle. 74. Prove Theorem VII. by first drawing a line through B parallel to A C. 75. Prove Theorem VII. by drawing a triangle upon the floor, walking over its perimeter, and turning at each vertex through an angle equal to the angle at that vertex. 76. Only one perpendicular can be drawn from a point to a straight line. (Two cases. 1st. When the point is without the line. 2d. When the point is within the line.) 77. Two straight lines perpendicular to a third are parallel. (13.) 78. If a line joining two parallels is bisected, any other line drawn through the point of bisection and joining the parallels is bisected. 79. If two triangles have two sides of one B respectively equal to two sides of the other, but the included angles unequal, the third side of the one having the included angle greater is greater than the third side of the other. (Place the triangles as in the figure; draw BE A bisecting the angle CBD, and join C and E) E 80. (Converse of 79.) If two triangles have two sides of one respectively equal to two sides of the other, but the third sides unequal, the included angle of the one having the third side greater is greater than the included angle of the other. (Prove it by proving any other supposition absurd.) 81. Prove in Theorem XIII. the angles of the two triangles equal by reference to (79); then that the triangles are equal by (40) or (41). . 82. (Converse of part of 62.) If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. 83. (Converse of part of 62.) If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. 84. (Converse of 63.) If a diagonal divides a quadrilateral into two equal triangles, is the figure necessarily a parallelogram? 85. The diagonals of a parallelogram bisect each other. 86. (Converse of 85.) If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 87. The diagonals of a rhombus bisect each other at right angles. 88. (Converse of 87.) If the diagonals of a quadrilateral bisect each other at right angles, the figure is a rhombus. 89. The diagonals of a rectangle are equal. 99. The diagonals of a rhombus bisect the angles of the rhombus. 91. Straight lines bisecting the adjacent angles of a parallelogram are perpendicular to each other. 92. From the vertices of a parallelogram measure equal distances the sides in order. The lines joining these points on the sides form a parallelogram. upon 93. Prove Theorem XX. by joining any point within to the vertices of the polygon. 95. If a pavement is to be laid with blocks of the same regular form, prove that their upper faces must be equilateral triangles, squares, or hexagons. (67.) (9.) · 96. If two kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are, 1st. Triangles and squares. How many of each must there be at each angular point? 97. If three kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are, 1st. Triangles, squares, and hexagons. 2d. Squares, hexagons, and dodecagons. How many of each must there be at each angular point? |