5. Construct two triangles, ABC, DEF, each with base 14 inch, and angles at the base 79° and 57°. It follows, from 4, that the remaining angles at A and D are equal, each being 44°. Putting the points of the dividers on A and B, and carrying the dividers, so adjusted, to DE, compare the magnitudes of AB and DE. In like manner compare the magnitudes of AC and DF. Next, cutting one of the triangles from the paper, place it upon the other. From this superposition what conclusion do you draw as to the areas of the triangles? Repeat the same construction, measurement, and superposition with the following triangles: Two whose bases are 13 in., and angles adjacent to base 38° and 110°. Two whose bases are 90 millimetres, and angles adjacent to base 89° and 57°. Two whose bases are 3 in., and angles adjacent to the base 49° and 95°. The result of our observations in all these cases is that if two triangles have their bases equal, and angles adjacent to the bases equal, the remaining angles are equal, and the sides opposite to equal angles are equal, and the areas are equal. In other words they are the same triangle in different positions. Another way of stating the fact is to say that if a side of a triangle and the angles adjacent to this side are fixed, then the remaining angle and sides are fixed, and area is fixed. 6. The following fact, demonstrated in Chapter VI., may be of service in connection with the succeeding exercises: The vertically opposite angles AEC and BED are equal; and also the vertically opposite angles AED and BEC. Exercises. A In numerical exercises, such as the first twelve, the teacher should solve the triangles by the usual trigonometrical formulæ, that he may inform the class as to the closeness of their approximations reached by instrumental methods. 1. The sides of a triangle are 35, 52 and 63 millimetres. Construct the triangle; and with the protractor measure the angles to the nearest degree. 2. The sides of a triangle are 36, 48 and 60 millimetres. Construct the triangle; and with the protractor measure the angles to the nearest degree. 3. The sides of a triangle are 66, 90 and 31 millimetres. Construct the triangle; and measure the angles to the nearest degree. 4. Two sides of a triangle are 21 and 11⁄2 inches, and the included angle is 47°. Construct the triangle; and measure the remaining side to the nearest sixteenth of an inch, and the remaining angles to the nearest degree. 5. Two sides of a triangle are 50 and 68 millimetres, and the included angle is 91°. Construct the triangle; and measure the remaining side to the nearest millimetre, and the remaining angles to the nearest degree. 6. Two sides of a triangle are 5 and 6 inches, and the included angle is 54°. Construct the triangle; and measure the remaining side to the nearest sixteenth of an inch, and the remaining angles to the nearest degree. 7. Two angles of a triangle are 55° and 65°, and the side adjacent to them is 27 millimetres. Construct the triangle; and measure the remaining angle to the nearest degree, and the remaining sides to the nearest millimetre. 8. Two angles of a triangle are 107° and 27°, and the side adjacent to them is 50 millimetres. Construct the triangle; and measure the remaining angle to the nearest degree, and the remaining sides to the nearest millimetre. 9. Two angles of a triangle are 53° and 66°, and the side adjacent to them is 4 inches. Construct the triangle; and measure the remaining angle to the nearest degree, and the remaining sides to the nearest sixteenth of an inch. 10. The sides of a triangle are 4, 6 and 7 inches. triangle; and measure the angles to the nearest degree. Construct the 11. Two sides of a triangle are 90 and 70 millimetres, and the included angle is 58°. Construct the triangle; and measure the remaining side to the nearest millimetre, and the remaining angles to the nearest degree. 12. Two angles of a triangle are 30° and 128°, and the side adjacent to them is 2 inches. Construct the triangle; and measure the remaining angle to the nearest degree, and the remaining sides to the nearest sixteenth of an inch. 13. Two lines AB and CD intersect in E, and, with the dividers, AE and EB are taken equal to one another, and also CE and ED equal to one another. Join AC, CB, BD, DA. What lines, angles and triangles are equal to one another? Give proof. 14. A triangle ABC is described, and on the other side of BC the triangle DBC is constructed with DB= AB and DC=AC. AD is joined. What lines, angles and triangles are equal to one another? Give proof. What angles are right angles? 15. A triangle ABC is described, and on the other side of BC the triangle DBC is constructed with DB= AC and DC=AB. AD is joined. What lines, angles and triangles are equal to one another? Give proof. 16. A triangle ABC is described, and on the same side of BC another triangle DBC is described with DB=AC and DC=AB. AD is joined. What lines, angles and triangles in the figure are equal? Give proof. If BA and CD be produced to meet in E, what are the triangles EAD, EBC? Give reasons. 17. From two lines diverging from A, equal lengths AB, AC are cut off, and also equal lengths AD, AE. CD, BE, BC and DE are joined. What lines, angles and triangles in the figure are equal? Give proof. 18. Two circles have the same centre O. AOB is a diameter of one, and COD a diameter of the other. AC and BD are joined. What lines, angles and triangles in the figure are equal? 19. Equal lines AB, AC are drawn, making equal angles with AE on opposite sides of it. At B and C equal angles ABF, ACG are constructed towards the same side. If AE, BF and CG be produced, will they hit the same point? Give proof. 20. Describe ABC, DBC, two isosceles triangles on the same base BC, but on opposite sides of it. How does AD divide the angles BAC, BDC? Give proof. 21. With centres A and B two circles are described, intersecting at C and D. How are the angles CAD, CBD divided by AB? How is CD divided by AB? What are the angles at the intersection of AB and CD? Give proof. 22. Construct an equilateral triangle ABC. At B and C construct equal angles GBC, GCB. Join AG. How does AG divide the angle BAC? Give proof. 23. With O as centre describe a circle, and, with the dividers, take three points on the circumference, A, B, C, such that the chords AB, BC are equal. How does OB divide the angles ABC, AOC? How does OB divide AC, and what are the angles at the point of intersection? Give proof. 24. ABC is any triangle. The side BC is produced to D, CA to E, and AB to F. How many degrees are there in the sum of the angles ACD, BAE, CBF? Verify by measurement and addition. |