PLANE GEOMETRY DEFINITIONS 1. Every material body occupies a limited portion of space. If we conceive the body to be removed, the space that is left, which is identical in form and magnitude with the body, is a geometrical solid. 2. A geometrical solid, then, is a limited portion of space. It has three dimensions: length, breadth, and thickness. 3. The boundaries of a solid are surfaces. A surface has but two dimensions: length and breadth. 4. The boundaries of a surface are lines. A line has length only. 5. The ends of a line are points. A point has position, but no magnitude. 6. A straight line is one that does not change its direction at any point. 7. A curved line changes its direction at every point. 8. A plane surface is a surface, such that a straight line joining any two of its points will lie wholly in the surface. 9. Any combination of points, lines, surfaces, or solids, is a geometrical figure. 10. A figure formed by points and lines in a plane is a plane figure. 11. Geometry is the science that treats of the properties, the construction, and the measurement of geometrical figures. 12. Plane Geometry treats of plane figures. 13. A plane angle is the amount of divergence of two lines that meet. The lines are the sides of the angle, and their point of meeting is the vertex. One way to indicate an angle is by the use of three letters. Thus, the angle in the accompanying figure is read angle ABC or angle CBA, the letter at the vertex being in the middle. If there is only one angle at the vertex B, it may be read angle B. B 3 A Another way is to place a small figure or letter within the angle near the vertex. The above angle may be read angle 3. The size of an angle in no way depends upon the length of its sides, and is not altered by either increasing or diminishing their length. 14. Two angles are equal if they can be made to coincide. Thus, angles ABC and DEF are equal, whatever may be the length of each side, if angle ABC can be placed B upon angle DEF so that the vertex B shall fall upon vertex E, BC fall upon EF, and BA fall upon ED. [It should be noticed that angle ABC can be made to coincide with angle DEF in another way, E i.e. ABC may be turned over and then placed upon DEF, BC falling upon ED, and BA upon EF.] 15. Two angles that have the same vertex and a common side are adjacent angles. Angles 1 and 2 are adjacent angles. 2 B A 16. If a straight line meets another straight line so as to make the adjacent angles that they form equal to each other, the angles formed are right angles. Angles ABC and ABD are right angles. In this case each line is perpendicular to the other. 17. An angle that is less than a right angle is acute, and one that is greater than a right angle is obtuse. An angle that is not a right angle is oblique. 18. A triangle is a portion of a plane bounded by three straight lines. The lines are called the sides of the triangle, and their angles the angles of the triangle. An equilateral triangle has three equal sides. B C An isosceles triangle has two equal sides. 19. A circle is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within, called the center. The bounding line is called the circumference. 20. The distance from the center to any point on the circumference is a radius. 21. Any portion of a circumference is an arc. Obtuse Radius B A A Acute B C Circumference 22. A theorem is a truth requiring demonstration. The statement of a theorem consists of two parts, the hypothesis and the conclusion. The hypothesis is that part which is assumed to be true; the conclusion is that which is to be proved. 23. A problem proposes to effect some geometrical construction, such as to draw some particular line, or to construct some required figure. 24. Theorems and problems are called propositions. 25. A corollary is a truth that may be readily deduced from one or more propositions. 26. A scholium is a remark made upon one or more propositions relating to their use, connection, limitation, or extension. 27. An axiom is a self-evident truth. AXIOMS 1. Things that are equal to the same thing are equal to each other. 2. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. If equals are multiplied by equals, the products are equal. 5. If equals are divided by equals, the quotients are equal. 6. If equals are added to unequals, the sums are unequal in the same order. 7. If equals are subtracted from unequals, the remainders are unequal in the same order. 8. If unequals are multiplied by positive equals, the prod ucts are unequal in the same order. 9. If unequals are divided by positive equals, the quotients are unequal in the same order. 10. If unequals are added to unequals, the greater to the greater, and the less to the less, the sums are unequal in the same order. 11. The whole is greater than any of its parts. 12. The whole is equal to the sum of all its parts. 13. Only one straight line can be drawn joining two points. [It follows from this axiom that two straight lines can intersect in only one point.] 14. The shortest distance from one point to another is measured on the straight line joining them. 15. Through a point only one line can be drawn parallel to another line. 16. Magnitudes that can be made to coincide with each other are equal. [This axiom affords the ultimate test of the equality of geometrical magnitudes. It implies that a figure can be taken from its position, without change of form or size, and placed upon another figure for the purpose of comparison.] Of the foregoing, the first twelve axioms are general in their nature, and the student has probably met with them before in his study of algebra. The last four are strictly geometrical axioms. 28. A postulate is a self-evident problem. POSTULATES 1. A straight line can be drawn joining two points. 2. A straight line can be prolonged to any length. 3. If two lines are unequal, the length of the smaller can be laid off on the larger. 4. A circumference can be described with any point as a center, and with a radius of any length. |