Geometry. 158. TRIANGLES-Continued. Right Triangle. 1. The sum of the angles of any triangle is equal to right angles or degrees. 2. In a right triangle there is one right angle. The other two angles are together equal to 3. In a certain right triangle one of the angles is an angle of 40°. How many degrees in each of the other two angles? Draw such a triangle. 4. Convince yourself by drawings and measurements that every equilateral triangle is equiangular. 5. Note that in every equiangular triangle each angle is one third of 2 right angles. So each angle is an angle of degrees. 6. If any one of the angles of a triangle is greater or less than 60, can the triangle be equiangular? Can it be equilateral? 7. If angle a of an isosceles triangle measures 50°, how many degrees in angle b? In angle c? a 159. Miscellaneous Review. 1. I am thinking of a right triangle one of whose angles measures 32°. Give the measurements of the other two angles. Draw such a triangle. 2. I am thinking of an isosceles triangle; the sum of its two equal angles is 100°. Give the measurement of its third angle. Draw such a triangle. 3. Let a equal the number of degrees in one angle of a triangle and b equal the number of degrees in another angle of the same triangle; then the number of degrees in the third angle is 180° − (a+b). If a equals 30, and b equals 45, how many degrees in the third angle? 4. Name three common multiples of 16 and 12. 5. Name the least common multiple of 16 and 12. 6. Find the sum of all the prime numbers from 101 to 127 inclusive. 7. Find the prime factors of 836. 8. With the prime factors of 836 in mind or represented on the blackboard, tell the following: (a) How many times is 19 contained in 836? (b) How many times is 11 x 19 contained in 836? (c) How many times is 19 x 11 x 2 contained in 836? DIVISIBILITY OF NUMBERS. 161. NUMBERS EXACTLY DIVISIBLE BY 2; BY 21; BY 31; BY 5; BY 10. 1. An integral number is exactly divisible by 2 if the right-hand figure is 0, or if the number expressed by its right-hand figure is exactly divisible by 2. EXPLANATORY NOTE.-Every integral number that may be expressed by two or more figures may be regarded as made up of a certain number of tens and a certain number (0 to 9) of primary units; thus, 485 is made up of 48 tens and 5 units; 4260 is made up of 426 tens and 0 units; 27562 is made up of 2756 tens and 2 units. But ten is exactly divisible by 2; so any number of tens, or any number of tens plus any number of twos, is exactly divisible by 2. 2. Tell which of the following are exactly divisible by 2, and why: 387, 5846, 2750, 2834. 3. Any number, integral or mixed, is exactly divisible by 2 if the part of the number expressed by figures to the right of the tens' figure, is exactly divisible by 21. 4. Show why the statement made in No. 3 is correct, employing the thought process given in the "Explanatory Note" above. 5. Tell which of the following are exactly divisible by 2, and why: 485, 470, 365, 4721⁄2, 38471⁄2. 6. Any number, integral or mixed, is exactly divisible. by 3 if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 3. 7. Tell which of the following are exactly divisible by 31, and why: 780, 283, 5763, 742, 80. Divisibility of Numbers. 8. Any integral number is exactly divisible by 5 if it right-hand figure is 0 or 5. Show why. 9. Any integral number is exactly divisible by 10 if its right-hand figure is 162. PROBLEMS. 1. How many times is 2 2. How many times is 2 3. How many times is 2 4.. How many times is 2 5. How many times is 3 6. How many times is 3 7. How many times is 3 contained in 5821?* 8. How many times is 5 contained in 3885? 9. How many times is 5 contained in 1260? 163. NUMBERS EXACTLY DIVISIBLE BY 25; BY 331; BY 12 BY 16; BY 20; BY 50. 1. Any integral number is exactly divisible by 25 if its two right-hand figures are zeros, or if the part of the number expressed by its two right-hand figures is exactly divisible by 25. EXPLANATORY NOTE.-Every integral number expressed by three or more figures may be regarded as made up of a certain number of hundreds and a certain number (0 to 99) of primary units; thus 4624 is made up of 46 hundreds and 24 units; 38425 is made up of 384 hundreds and 25 units; 8400 is made up of 84 hundreds and 0 units. But a hundred is exactly divisible by 25; so any number of hundreds, or any number of hundreds plus any number of 25's is exactly divisible by 25. *2 is contained in 582 (4 × 58) + 1 times. Why? +3 is contained in 7863 (3 × 78) + 2 times. Why? Divisibility of Numbers. 2. Tell which of the following are exactly divisible by 25, and why: 37625, 34836, 27950, 38575. 3. Every number, integral or mixed, is exactly divisible by 33, if that part of the number expressed by the figures to the right of the hundreds' figure is exactly divisible by 331. 4. Show why the statement made in No. 3 is correct, employing the thought process given in the "Explanatory Note" under No. 1 on the preceding page. 5. Tell which of the following are exactly divisible by 331, and why: 364663, 2375, 468331, 38900, 46820. 6. Any number, integral or mixed, is exactly divisible by 12, if the part of the number expressed by the figures to the right of the hundreds' figure, is exactly divisible by 121. Show why. 7. Tell which of the following are exactly divisible by 12, and why: 375, 8371, 6450, 4329, 74671. 8. Any number, integral or mixed, is exactly divisible by 163, if 9. Tell which of the following are exactly divisible by 16: 46331, 5460, 2350, 37400, 275831, 254163. 10. Any integral number is exactly divisible by 20 if the number expressed by its two right-hand figures is exactly divisible by 20. Show why. 11. Tell which of the following are exactly divisible by 20, and why: 3740, 2650, 3860, 29480, 3470. 12. Tell which of the following are exactly divisible by 50, and why: 2460, 3450, 6800, 27380, 25450. |