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16 THE FUNDAMENTAL PRINCI- PLES OF GEOMETRY. If the equatorial circumference of the earth is 25,000 miles, how many miles apart are two places on the equator, the distance between them being 20°? 20°=1/18 circle. Ans. 1,388.8 miles, nearly. What is the length of a degree on a circle whose diameter is 18 ft.? The 60th parallel of latitude is a cir- cle one-half as long as the equator. How many miles due east of Christiania is St. Petersburg, both situated on this par- allel, the former being 10° east of Green- wich, and the latter 30°. e«astr-. Ans: 691,160 statute miles. 1° on the circumference of a circle is 5 in. What is the length of the circum- ference? Ans. 150. ft; The circumference of a circle is 9,000 ft. What is the length of 1°, 1'? Ans. 25 tt; 4 ft., nearly. How many degrees are there between XII and the I on the face of a clock? Ans. 15°; between XII and VI? Ans. 90°; between the XII and Tilr Ans. 45°. If one degree of the earth's circum- ference is 69 1/6 statute miles, find the length of the circumference. Through how many degrees does the minute hand of a clock pass in 1 hour? in 15 minutes? in 5 minutes? in 10 min- utes? in 1 minute? in 3 minutes? How angles are measured: When the ends. of two straight lines meet they form an angle. On the com- pass a line from the center drawn through North and NE forms an angle of 4 points or 45°. 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 es UTS, the letter at the ver- tex being placed between the two others. _ The use of the three letters is necessary where two or : more angles have vertices at the same point, as in the accompanying figure, where UX, VX,and WX meet at the point X. The angle at the center of a circle has the same number of degrees as the arc between the sides of the angle. Thus, in the following figure the angle AOF has the same number of degrees or points as the arc AF, or the arc ABC, or the arc CD, 1. How many degrees in the angle 7, if S contains v w ae ee, V. measures 110°. How many degrees does U meas- ure? v If one of two supplemen- tary angles measures 6314°, how many degrees are there in the other angle? TRAE MARINE KEVIEW How many degrees are there in-an an- gle supplementary to one of 47° 45'? ' 2. Construct angle 5, 60°; angle 4, 50°. Measure angle 3. How many degrees and dee minutes will there be in an- gle 5 when 3 contains 4914 and 4 con- tains 8334°? When angle 3 contains 36° 30' and an- gle 5 contains 79° 45', how many degrees and minutes will angle 4 contain? 3. Erect a perpendicular at each ex- tremity of a horizontal line. At each ex- tremity of a vertical line. At each ex- tremity of an oblique line. Note.--A line making a right angle with another line is said to be perpendicular to it 4. Construct a square upon a horizontal line. Upon an oblique line. 5. Draw two lines intersecting at au angle of 100°. Mark in each of the other three angles the Sc number of degrees it contains. 6. Draw two lines making an angle (6) of 150°. Con- S28 struct an adjacent angle (7) : : containing 80°. How many degrees will angle 8 contain? 7. How many degrees will there be in the sum of five angles having the same ver- tex? G The circumference of this circle is di- vided into 36 equal parts. How many degrees are there in each part? How many degrees are there in each of the following angles? Wop, BOC, COD DOL, EOF, FOG, GOA, AOE, DOF. 8. Draw a line, AB, meeting a horizon- tal line; BC, at an angle of : 58°. Draw a third line, DE, . : parallel to the horizontal eee line, and cutting the oblique d line. What angles does it make with the oblique line? Draw a fourth line, FG, parallel to the oblique line, and cutting both horizontal lines. Mark in each of the 12 angles the number of degrees it contains. 9. OR and UV are parallel lines, cut by a line ST. If the angle b measures 50°, how many Ae ad degrees does a measure? as Find the number of de- : grees in each of the other six angles. TRIANGLES. 10. Construct a triangle, KLM, mak- ing the angles at the base 28° ee and 120°, respectively. Draw be NO, parallel to LM. ee Is the angle e equal to any angle of the triangle? How many degrees does it contain? Is the angle f equal to any angle of the triangle? How many degrees does it contain? How many degrees are there in the sum of the angles e, g, and f? How many degrees are there in the angle g? 11. How many degrees are there in the three angles of any triangle? 12. Two angles of a triangle measure 36° and 65°, respectively. How many degrees does the third angle contain? 13. Draw a triangle containing two an- gles of 50° and 70°, respectively. How many degrees are there in the third angle? Measure each side, and mark on the side its length. Opposite which angle is found the long- est side? Opposite which, the shortest side? 14. Draw a triangle having two angles of 75° each. Are any two of its sides equal? Draw a triangle having two angles of 50° each. Are any of its sides equal? 15. Draw a triangle having two angles of 60° each. How many degrees does the third angle contain? Are any of its sides equal? 16. If a triangle has two of its sides equal, what is true of its angles? 17. If a triangle-has three of its sides equal, what is true of its angles? A triangle having all its sides equal, is called an equilateral triangle. A triangle having two equal sides, is called an isosceles triangle. A triangle having all its sides unequal, is called a scalene triangle. 18. How does a perpendicular let fall upon the base of an isosceles A triangle from the opposite angle divide the angle? How does it divide the base? How ¢ do the angles at the base of an isosceles triangle compare with each other as to size? ' c 8 The unequal side of an isosceles tri- angle is called the base, 19. Draw an isosceles triangle having the base a vertical line. An isosceles triangle having the vertex below the base. One having an oblique line for the base. Draw a right-angled isosceles triangle. id a NN i aie aa aa