53. Postulates. The following are among the most important postulates used in geometry. Others will be introduced as needed. 1. One straight line and only one can be drawn through two given points. 2. A straight line may be produced to any required length. To produce AB means to extend it through B; A to produce BA means to extend it through A. 3. A straight line is the shortest path between two points. 4. A circle may be described with any given point as a center and any given line as a radius. 5. Any figure may be moved from one place to another without altering its size or shape. 6. All straight angles are equal. 54. COROLLARY 1. Two points determine a straight line. This is only a brief way of stating Postulate 1. 55. COROLLARY 2. Two straight lines can intersect in only one point. For if they had two points in common they would coincide (Post. 1). 56. COROLLARY 3. All right angles are equal. For all straight angles are equal (Post. 6), and a straight angle (§ 34) is twice a right angle. Hence Axiom 4 applies. 57. COROLLARY 4. From a given point in a given line only one perpendicular can be drawn to the line. For if there could be two perpendiculars to DA at O, as OB and OC, we should have angles AOB and AOC both right angles, which is impossible (§ 56). D 58. COROLLARY 5. Equal angles have equal complements, equal supplements, and equal conjugates. 59. COROLLARY 6. The greater of two angles has the less complement, the less supplement, and the less conjugate. EXERCISE 4 1. If 10° + <x=27° 30', find the value of x. Find the value of x in each of the following equations: 19. If x+20° = y and y — 5° = 2x, what is the value of x and of y? Find the value of x and of y in each of the following sets of equations: 26. If <10° and y=7° 30′, what can be said as to the value of x+y? 27. In Ex. 26, what can be said as to the value of x y? BOOK I RECTILINEAR FIGURES PROPOSITION I. THEOREM 60. If two lines intersect, the vertical angles are equal. Given the lines AC and BD intersecting at 0. (The two adjacent angles which one straight line makes with another are together equal to a straight angle.) BOC+ZCOD: a st. Z. § 43 Likewise § 43 (If equals are subtracted from equals the remainders are equal.) Q.E.D. 61. Nature of a Proof. From Prop. I it is seen that a theorem has (1) certain things given; (2) a definite thing to be proved; (3) a proof, consisting of definite statements, each supported by the authority of a definition, an axiom, a postulate, or some proposition previously proved. 62. Triangles classified as to Sides. A triangle is said to be scalene when no two of its sides are equal; isosceles when two of its sides are equal; Δ Scalene Isosceles Equilateral 63. Triangles classified as to Angles. A triangle is said to be right when one of its angles is a right angle; obtuse when one of its angles is an obtuse angle; acute when all of its angles are acute angles; equiangular when all of its angles are equal. AA Right Obtuse Acute Equiangular 64. Corresponding Angles and Sides. If two triangles have the angles of the one respectively equal to the angles of the other, the equal angles are called corresponding angles, and the sides opposite these angles are called corresponding sides. Corresponding parts are also called homologous parts. 65. Square. A rectilinear figure having four equal sides and four right angles is called a square. 66. Congruent. If two figures can be made to coincide in all their parts, they are said to be congruent. 67. COROLLARY. Corresponding parts of congruent figures are equal. When equal figures are necessarily congruent, as in the case of angles or straight lines, the word equal is used. For symbols see page vi. PROPOSITION II. THEOREM 68. Two triangles are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other. Given the triangles ABC and XYZ, with AB equal to XY, AC equal to XZ, and the angle A equal to the angle X. To prove that ▲ ABC is congruent to ▲ XYZ. Proof. Place the ▲ ABC upon the AXYZ so that A shall fall on X and AB shall fall along XY. Post. 5 (Any figure may be moved from one place to another without altering its size or shape.) Then B will fall on Y, (For AB is given equal to XY.) AC will fall along XZ, and C will fall on Z. (For AC is given equal to XZ.) .. CB will coincide with ZY. Post. 1 (One straight line and only one can be drawn through two given points.) .. the two coincide and are congruent, by § 66. Q.E.D. 69. COROLLARY. Two right triangles are congruent if the sides of the right angles are equal respectively. The right angles are equal (§ 56). How does Prop. II apply ? |