We define a plane figure as a figure formed by points and lines all lying in the same plane. A figure is called rectilinear when it is composed of straight lines only. A straight line is sometimes called a right line. 9. Geometry treats of the properties, construction, and measurement of geometrical figures. Plane Geometry treats of plane figures only. Solid Geometry treats of figures which are not plane. 10. We define an angle as the figure formed by two straight lines drawn from a point. The point is called the vertex of the angle, and the straight lines its sides. α B -C 11. If there is but one angle at a given vertex, we designate it by the letter at the vertex. But if two or more angles have the same vertex, we avoid ambiguity by naming also a letter on each side, reading the vertex letter between the others. Thus, we should read the angle of § 10 "angle B"; but if there were other angles at the same vertex, we should read it either ABC or CBA. Another way of designating an angle is by means of a letter placed between its sides; thus, we may read the angle of § 10 "angle a." 12. Two figures are said to be equal when one can be applied to the other so that they shall coincide throughout. To prove two angles equal, we do not consider the lengths of their sides. B A D CE F Thus, if angle ABC can be applied to angle DEF in such a manner that point B shall fall on point E, side AB on side DE, and side BC on side EF, the angles are equal, even if side AB is not equal in length to side DE, and BC to EF. 13. We call two angles adjacent when they have the samə vertex, and a common side between them; as AOB and BOC. We call angle AOC the sum of angles AOB and BOC. We also regard angle AOC as greater than 0angle BOC, and angle BOC as less than angle AOC. B C 14. If from a point in a straight line a line be drawn in such a way as to make the adjacent angles equal, each of the adjacent angles is called a right angle, and the lines are said to be perpendicular to each other. Thus, if from point A in line CD line AB be drawn in such a way as to make angles BAC and BAD equal, each of these angles is a right angle, and AB and CD are perpendicular to each other. B D A 15. Let C be any point in straight line AB; and let straight line CD be drawn in such a way as to make angle BCD less If CE is this position, by the definition of § 14, CE is perpendicular to AB at C. Then, at a given point in a straight line, a perpendicular to the line can be drawn, and but one. The following are immediate consequences of § 15: 16. Any two right angles are equal. 17. If two adjacent angles have their exterior sides in the same straight line, their sum is equal to two right angles. For in figure of § 15, the sum of angles ACD and BCD equals the sum of angles ACE and BCE, or two right angles. 18. The sum of all the angles on the same side of a straight line at a given point is equal to two right angles. 19. The sum of all the angles about a point in a plane is equal to four right angles. For if a side of any angle, as ОA, be extended to E, the sum of the angles on either side of straight line AE is, by § 18, equal to two right angles. E B A 20. If the sum of two adjacent angles is equal to two right angles, their exterior sides lie in the same straight line. If the sum of angles ACD and BCD is two right angles, side AC if extended. through C must coincide with CB; for if it did not the sum of the angles would be A less or greater than 180° (§ 17). 21. We define a triangle as a portion of a plane bounded by three straight lines; as ABC. We call the lines AB, BC, and CA the sides of the triangle; and their intersections, A, B, and C, the vertices. We define the angles of the triangle as the angles CAB, ABC, and BCA, between the adjacent sides. B 22. A triangle is called scalene when no two sides are equal; isosceles when two sides are equal; equilateral when all its sides are equal; equiangular when all its angles are equal. Scalene Isosceles Equilateral A right triangle is a triangle which has a right angle; as ABC, which has a right angle at C. The side AB opposite the right angle is called the hypotenuse, and the other sides, AC and BC, the legs. 23. We define a circle as a portion of a plane bounded by a curve, called the circumference, all points of which are equally distant from a point within A called the centre. An arc is any portion of the circum ference; as AB. B B Q A Q A radius is a straight line drawn from the centre to the circumference; as OA. CONSTRUCTIONS 24. At a given point in a straight line, to draw a perpendicu Let it be required to draw a line perpendicular to the straight line AB, at the point C. With C as centre, and any straight line less than AC as radius, describe arcs intersecting AC at D, and BC at E. With points D and E as centres, and any radius, describe arcs intersecting at F. Then, the straight line drawn from C through F will be perpendicular to AB at C. The reason for the above construction will be found in § 58. 25. From a given point without a straight line to draw a perpendicular to the line. Let it be required to draw through any point C without the straight line AB a line perpendicular to AB. With Cas a centre, and any radius, describe an arc cutting AB at points D and E. With points D and E as centres, and another radius, describe arcs intersecting at F. Then, the straight line CG drawn from C through F will be perpendicular to AB. The reason for the construction will be found in § 58. 26. To bisect a given straight line. Let it be required to divide the straight line AB into two equal parts. With points A and B as centres, and with the same radius, describe arcs intersecting at C and D. Draw straight line CD intersecting AB at E. Then, E is the middle point of AB. The reason for the construction will be found in § 58. |