CHAPTER XVI. ELEMENTARY GEOMETRY. PROBLEMS IN CONSTRUC TION. PRACTICAL APPLICATIONS. CALCULATION OF HEIGHTS AND DISTANCES. - MENSURATION. ELEMENTARY GEOMETRY. 1251. Angles. When two straight lines meet at a point, they are said to form an angle. The point at which the lines meet is called the vertex of the angle. When two angles are formed by the meeting of two straight lines, they are called adjacent angles. A and B are adjacent angles. C and D are adjacent angles. B The angles E and G, formed by the intersection of two straight lines, are called vertical, or opposite angles. Fand H are vertical angles. E and F, F and G, G and H, H and E, are adjacent angles. When two adjacent angles are equal to each other, each is said to be a right angle. The angles I, J, K, L, M, N, O, P are right angles. An angle that is smaller than a right angle is called an acute angle; one larger than a right angle is called an obtuse angle. Q is an acute angle; R is an obtuse angle. K L R کا H Angles that are not right angles are called, without regard to their size, oblique angles. 1252. Designation of Angles. The angle formed by the lines ST and TU may be called the angle T. It is frequently better to call it the angle STU or UTS, the letter at the vertex being placed between the two others. The use of the three letters is necessary where there is more than one angle having its vertex at the same point, as in the accompanying figure, where UX, VX, and WX meet at the point X. 1253. Measurement of Angles. An angle is measured by the arc of a circle, the center of the circle being at the vertex of the angle. The angle 1 2 3 is measured by the arc YZ; the angle 456 by the arc ab. 1254. Circular Measure. 6 b 5 S T U X -W 1255. The number of degrees in an arc or an angle is determined by a protractor. D To measure an angle, XYZ, for instance, produce the lines YX and YZ. Place the point A of the pro Using the Y X Z tractor on the vertex (Y) of the angle, and the edge AC on the line YZ produced. lower line of figures, read off from the protractor the number of degrees at the point where the line YX produced cuts the semi-circle. In measuring the angle DEF, the line AB is placed on EF, the point A on the vertex E. The number of degrees in this case is read from the upper row of figures. EXERCISES IN CONSTRUCTION. D F E 1256. NOTE. - In the following 100 exercises, the ruler, the compasses, and the protractor may be used. The drawing should be carefully done with a sharp, hard pencil. 1. Draw an obtuse angle formed by two lines, each one inch long. Draw an acute angle formed by two lines, each six inches long. Which is the larger? 2. Fold a piece of paper twice, so that the lines made by the creases will form four right angles. Fold a piece of paper so as to make four angles that are not right angles. 3. The lines GH and IJ intersect at K, making four right angles. Which arc is longer, 78, or cd? Which contains the greater number of degrees? I H K G 4. Draw two lines meeting at an angle of 45°. Two lines meeting at an angle of 90°. Two meeting at an angle of 135°. 5. Draw two lines making two angles, one of which measures 60°. How many degrees does the other angle contain? 60° 1257. NOTE. is called a vertical line; one parallel to the top or the bottom of the paper is called a horizontal line; others are called oblique lines. A line parallel to the right or the left side of the paper KL is a vertical line, MN is a horizontal line, OP and QR are oblique lines. K M N R 0 6. To a horizontal line draw a line making two equal adjacent angles. How many degrees does each angle contain? To a vertical line draw a line making two equal adjacent angles. How many degrees does each angle contain? To an oblique line draw a line making two equal adjacent angles. How many degrees does each angle contain? 7. How many degrees are there in a right angle? 8. To an oblique line draw a line making two unequal adjacent angles. How many degrees are there in the sum of the two angles? 9. How many degrees in the angle T, if S contains 75°? V measures 110°. How many degrees does U measure? If one of two adjacent angles measures 63°, how many degrees are there in the other angle? T S How many degrees are there in an angle adjacent to one of 47° 45'? 10. Construct angle 5, 60°; angle 4, 50°. Measure angle 3. How many degrees and minutes will there be in angle 5, when 3 contains 491° and 4 contains 833°? 3 When angle 3 contains 36° 30' and angle 5 contains 79° 45', how many degrees and minutes will angle 4 contain? 11. Erect a perpendicular at each extremity of a horizontal line. At each extremity of a vertical line. At each extremity of an oblique line. 1258. NOTE.-A line making a right angle with another line is said to be perpendicular to it. 12. Construct a square upon a horizontal line. Upon an oblique line. 13. Draw two lines intersecting at an angle of 100°. Mark in each of the other three angles the number of degrees it contains. 100 14. If one of the four angles formed by two intersecting lines measures 90°, what does each of the other three measure? If one measures 60°, what does each of the others measure? 15. At each extremity of a horizontal line draw a line making an angle of 40° with the first line. 16. At each extremity of a vertical line draw a line making an angle of 100° with the first line. 17. At one extremity of an oblique line draw a line making with the first line an acute angle; at the other extremity draw a line making an obtuse angle with the first line. 18. Draw two lines making an angle (6) of 150°. Construct an adjoining angle (7) containing 80°. How many degrees will angle 8 contain? 19. How many degrees will there be in the sum of five angles having the same vertex? 20. Draw five equal angles having a common. vertex. Is any one of these five angles adjacent to any other? 6 8 21. Draw six equal angles having a common vertex. Is any angle adjacent to the angle next it? Why? Are any of the angles vertical? Why? 22. Draw two angles, one of 65° and the other of 25°. Draw a third angle equal to the sum of both. Draw an angle equal to their difference. |