The foregoing construction may be made by use of a draftsman's triangle, instead of compasses, as follows: Place the triangle with one side along the given line AB. Then adjust a ruler against another side of the triangle. Holding the ruler in this position, slide the triangle along it until the side that was along AB comes to the given point P. Then draw the line CD along this side. Show that this construction is equivalent to the foregoing. EXERCISES 1. By use of compasses and straightedge, draw a line through a given point parallel to a given line. Repeat the process until it is thoroughly mastered. 2. By use of a triangle and ruler, draw a line through a given point parallel to a given line. Repeat the construction until proficient. 3. Make a large triangle of wood for use in drawing parallel lines on the blackboard. Practice using it at the blackboard. 22227 4. The Greek design above is called a meander. Make a copy of it. How are the parallel lines drawn? 5. In the figure there are five parallel diagonal lines. Do they look parallel? Construct a large figure like this by first drawing the five parallel diagonal lines, and see if your figure has the same deceptive appearance. 44. Theorem. - Two straight lines parallel to the same straight line are parallel to each other. Suggestions. Draw a transversal, forming corresponding angles Z1, Z2, Z3, as in the figure. AB will be parallel to CD if what can be shown? The latter will be true if it can first be shown that 21=23 and 22 = 23. Why? Hence begin by proving these equalities. Write the proof in full. 45. A corollary. - A corollary is a statement of a truth that is easily deduced from another truth. The following is a corollary of the theorem in § 44. 46. Corollary. - Only one straight line can be drawn through a given point parallel to a given straight line. For, if more than one could be drawn, they would be parallel by § 44, which is not possible because intersecting lines are not parallel. 47. A triangle. - If three points not in a straight line are connected by three line-segments, the figure formed is a triangle. The three points are called the vertices of the triangle, and the line-segments are called the sides. A triangle is named by naming its vertices. The word "triangle" is represented in written work by the symbol Δ. Thus the triangle above is named "triangle ABC," and may be written △ ABC. The angle MEF, formed by the side EF and a prolongation of the side DE through E, is called an exterior angle of triangle DEF. The angles formed by the sides themselves are called the angles, or, for distinction, the interior angles, of the triangle. 48. Theorem. - The sum of the three angles of any triangle is equal to a straight angle. Hypothesis. ∠ A, ∠ B, and ∠ Care the angles of ABC. Proof. 1. Produce AB through B to D, and draw BE || AC, forming < 1 and ∠2, as in the figure. Ax. XII 5... ∠ A + ∠ B + ∠C= st. Z. Draw a figure, and write out the complete proof. NOTE. -Pythagoras, the famous Greek philosopher and mathematician, knew this theorem about 500 в.с. It is one of the most important and prac tical theorems of geometry. 49. Corollary 1. - If two triangles have two pairs of equal angles, the third pair of angles must be equal. The proof is left to the student. 50. Corollary 2. - In any triangle there must be at least two acute angles. The proof is left to the student. 51. Corollary 3. — An exterior angle of any triangle is equal to the sum of the two opposite interior angles, and hence is greater than either one of them alone. SUGGESTION. - Prove ∠ DBC = ∠ A + ∠ C. Write the proof in full. EXERCISES 1-4. Prove the theorem in §48 by the methods suggested by the 5. If the three angles of a triangle are equal, show that each con tains 60°. 6. If one angle of a triangle is 37° 40′ and the other two angles are equal, find the size of each of the equal angles. 7. If one angle of a triangle is 64° 15' and another is 45° 30′ 30′′, find the third angle. 8. If one angle of a triangle is a right angle, what is the sum of the other two angles? Explain. 9. Show that if one angle of a triangle is a right angle, the other two angles must be complementary. 10. Of the three angles of a triangle the second is twice as large as the first and the third three times as large as the first. How many degrees are there in each? (Form an equation.) 11. If one angle of a triangle is a right angle, and one of the acute angles is four times the other, how many degrees are there in each? 12. In a certain triangle an exterior angle is twice the adjacent interior angle, and the two opposite interior angles are equal. How many degrees are there in each of the angles of the triangle? 13. Given two angles of a triangle, with compasses and straightedge construct the third angle. Base the construction on § 48. 14. In this figure Za equals the sum of what two angles? b? c? From these equations find the number of degrees in Za + 2b + ∠c. 15. Prove that if AD bisects ∠A of ∆ABC, and BD bisects ∠EBC, the exterior angle, then ∠ D = } ∠ C. SUGGESTION. - Express the sum of the angles of ∆ABC in terms of ∠ D, ∠ A, ∠ B, and ∠C. Set this equal to ∠A + ∠B + ∠C. Why? A C C 6 1 3a D B E 16. I may test my accuracy in measuring angles with a surveyor's instrument, as follows: I drive stakes in the ground at three points A, B, and C, so that they are not in a straight line. I set the instrument over A and measure ∠BAC. Then I set it over B and measure ∠ABC. Then I set it over Cand measure ∠ACB. The angles are found to be 46° 12', 32° 25', and 101° 33′. Am I accurate? NOTE. - It would be found practicable and interesting to make an instrument for measuring angles, such as that shown in the drawing, and to use it out of doors in measurements like those suggested in the above problem. The instrument would prove useful for many other out-of-door problems. It consists of a board (a drawing board is convenient) mounted upon a tripod. On the board is a circle marked off into 360°. A pointer, which is used for sighting to objects as well as for turning off angles, swings on a pivot at the center of the circle. |