d. The difference of two straight lines. e. Two adjacent angles. f. An angle doubled. g. An angle bisected. h. The sum of two angles. i. The difference of two angles. j. Two perpendicular lines. k. Two oblique lines. 2. Draw two intersecting straight lines. Name pairs of adjacent angles. Why are they adjacent? 3. Name pairs of supplementary angles. Why are they supplementary? 4. Make the definition of supplementary adjacent angles. Are supplementary angles always adjacent? 5. Name two angles that are supplements of the same angle. What can be said of these two angles? 6. Name pairs of vertical angles. Why are they vertical? 7. How many degrees in an angle that is 1⁄2 of a right angle? of a right angle? % of a straight angle? % of a right angle? % of two right angles? 8. What is the complement of 15° ? 9. What is the supplement of 60° ? 60° 30'? 89° 30′ 15′′ ? 120° 20′ 20′′ ? CHAPTER II EQUAL ANGLES The basic method is: All straight angles are equal. (23.) 30. All right angles are equal. 31. Complements of the same or equal angles are equal. 32. Supplements of the same or equal angles are equal. *33. If two straight lines intersect, the vertical angles are equal. 34. Define Corresponding Angles; Alternate Interior Angles. If two straight lines are crossed by a transversal making the corresponding angles equal, the lines are parallel. 36. If two parallel lines are crossed by a transversal, the corresponding angles are equal. *37. If two parallel lines are crossed by a transversal, the alternate interior angles are equal. *38. If two straight lines are crossed by a transversal making the alternate interior angles equal, the lines are parallel. SUMMARY 39. State seven methods of proving angles equal. State two methods of proving lines parallel. 40. EXERCISES 1. Draw two parallel lines crossed by a third straight line. Name pairs of equal angles. Why are they equal? 2. In the figure of ex. 1, if one angle equals 60°, what is each of the other three? 3. In the same figure, if one angle is 119° 30', what is each of the other three? 4. If an angle equals its complement, how many degrees are in the angle? NOTE. Let x = the angle; then x = complement; then x+x= ? 5. If an angle equals one-half its complement, how many degrees in the angle? 6. If an angle equals twice its complement, how many degrees in the angle? 7. If an angle equals its supplement, how many degrees in the angle? 8. If an angle equals one-half its supplement, find the angle. If an angle equals twice its supplement, find the angle. 9. 10. Of two complementary angles, one is five times the other. How many degrees in each? 11. Of two supplementary angles, one is five-sevenths of the other. Find each angle. 12. If the sum of three equal angles equals two right angles, how many degrees in each? 13. Divide a line, 84 units in length, into two parts in the ratio 4 to 3. NOTE. Let 4x be one part and 3x be the other. 14. Divide a line, 121 units in length, into two parts in the ratio 7 to 4. = = 15. In the figure of article 33, angle 1 + angle 2: ? Angle 3 + angle 4? Angle 1+ angle 2 + angle 3 + angle 4: ? Therefore, the sum of all the angles about a point in a plane equals what? 16. If the angular space about a point in a plane is divided into six equal parts, how many degrees in each part? 17. What is the test for congruency in all figures? Are straight lines or angles congruent if they are equal? Is this true of all figures? Are all figures equal if they are congruent? CHAPTER III ANGLE-SUMS The basic method is: If one straight line meets another straight line, the sum of the two adjacent angles equals two right angles. (Art. 26.) *41. If two parallel lines are cut by a transversal, the sum of the two interior angles on the same side of the transversal equals two right angles. *42. The sum of the angles of a triangle equals two right angles. *43. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. 44. If two angles of one triangle equal two angles of another triangle, the third angles are equal. 45. The sum of the angles of a quadrilateral equals four right angles. 46. Define Polygon; Regular Polygon. 47. Into how many triangles can a polygon of n sides be divided by drawing diagonals from any one vertex? *48. The sum of the interior angles of a polygon is equal to two right angles multiplied by a number two less than the number of sides. *49. The sum of the exterior angles of a polygon, formed by producing each side in succession, is equal to four right angles. SUMMARY 50. State the basic theorem of angle-sums. State the commonest theorem. State the formula for the sum of the interior angles of a polygon; for an angle of a regular polygon. 51. EXERCISES 1. The sum of the two acute angles of a right triangle equals one right angle. 2. Each angle of an equiangular triangle is one-third of two right angles. 3. Construct an angle of a triangle, having given the other two. 4. Construct an acute angle of a right triangle, having given the other acute angle. 5. Construct an angle of: 45°; 60°; 30°; 22°30′; 82°30′; 7°30′; 3°45′. 6. The bisectors of a pair of vertical angles lie in the same straight line. 7. If an angle of a triangle is 60° and two of the angles are equal, the other angles are what? 8. Find the sum of the interior angles of a polygon of five sides; six sides; eight sides; twenty sides. 9. The sum of the interior angles of a polygon is 16 right angles. How many sides has the polygon? 10. How many degrees in each angle of a regular polygon of twelve sides? 11. Each angle of an equiangular polygon is 140°. How many sides has the polygon? 12. How many sides has a polygon, the sum of whose interior angles equals twice the sum of the exterior angles? 13. Find the sum of the interior angles of a polygon of 52 sides; 77 sides; 1,002 sides; 4 sides; 3 sides; 2 sides. 14. How many sides has a polygon the sum of whose interior angles is 10 right ? 12 right ? 14 right ? 96 right ? 15. How many sides has a polygon if the sum of the interior angles equals the sum of the exterior angles? 16. How many degrees in one angle of a regular polygon of 5 sides? 6 sides? 8 sides? 10 sides? 17. If the base angle of an isosceles triangle is two-fifths of two right angles, the bisector of that angle divides the triangle into two isosceles triangles. |