What position do you observe the greatest, intermediate and least angles occupy with respect to the greatest, intermediate and least sides respectively? We shall find for all triangles a definite answer to the preceding question in the following proof: Let AC be greater than AB, and let AD A be equal to AB. Then (Ch. II., 2) the angles ABD and ADB are equal. But the angle ABC is greater than B ABD, and the angle ACB is less than ADB (Ch. V., 3). Therefore the angle ABC is greater than the angle ACB. That is, in any triangle, the greater side has the greater angle opposite it. 5. Construct a triangle with angles 40°, 60°, 80°, and, using the dividers or compasses, arrange the sides in order of magnitude. Make the same examination in the case of a triangle whose angles are 100°, 50°, 30°. What position do you observe the greatest, intermediate and least sides occupy with respect to the greatest, intermediate and least angles respectively ? We shall find for all triangles a definite answer to the preceding question in the following proof: Let the angle ABC be greater than the angle A ACB; then the side AC is greater than the side AB. For, with bevel or protrac , tor, construct the angle CBD equal to the angle ACB, so that DBC is an isosceles triangle. Then AC AD + DC AD + DB > AB, the straight line AB being the shortest distance between A and B. Hence in any triangle the greater angle has the greater side opposite to it. B Exercises. 1. Construct a quadrilateral figure. With the protractor measure the number of degrees in each of the angles, and add them. What is the sum ? Deduce this sum also from geometrical truths already reached. 2. Produce the sides of the quadrilateral, and measure the exterior angles. What is their sum? Deduce this also from knowing the sum of the interior angles. 3. Construct a polygon with any number of sides, ABCDE Taking the sides in order, produce each from the preceding angle, as in the figure of 2, Ch. V. Placing your pencil along AB, turn it through the exterior angle at B into coincidence with BC; then through the exterior angle at C; and so on, until it has been turned through all the exterior angles. How much has the pencil been turned? What, therefore, do you conclude the sum of all the exterior angles of any-polygon is? Verify this by measurement with protractor. 4. From the result reached in the previous question, show that all the interior angles of any polygon are equal to twice as many right angles as the figure has angles (or sides), less four right angles. 5. How many right angles is the sum of all the angles in a pentagon (5 sides) equal to? If the angles be equal, how many degrees àre there in each ? 6. How many right angles is the sum of all the angles in a hexagon (6 sides) equal to? If the angles be equal, how many degrees are there in each ? 7. Construct an isosceles triangle ABC (AB=AC). In AB take any point D. With the dividers or compasses determine whether D is nearer to B or to C. Give reason. 8. ABCD is a right-angled equilateral four-sided figure. AC is joined. Any point E is taken within the triangle ABC. Is E nearer to B or to D? Give reasons. 9. A triangle can have only one angle either equal to or greater than a right angle, i.e., at least two of the angles of a triangle must always be acute angles. 10. The perpendicular is the least line that can be drawn from a given point to a given line; and any line nearer to the perpendicular is less than one more remote. 11. ABCD is a four-sided figure. How does the sum of the exterior angles at A and C compare in magnitude with either of the interior angles B or D? Give reasons. 12. ABC is a triangle, and O is a point within it. Is the angle BOC greater than, equal to, or less than the angle BAC? Give reasons. 13. Can more than two equal straight lines be drawn to a straight line from a point without it? Give reasons. 14. Use the result obtained in the previous question to show that a circle cannot cut a straight line in more than two points. 15. In a right-angled triangle the hypotenuse is the greatest side. 16. In the triangle ABC can you find a point D, such that AD is equal to or greater than the greater of the sides AB, AC? 17. In any triangle can you find a point such that the distance from it to any one of the angles is equal to or greater than the greatest of the sides ? 18. Describe two circles with the same centre, i.e., concentric. Take a point A on the circumference of one, and a point B on the circumference of the other. When will the line AB be least? Give reasons. 19. A, B, C are three points on a line, at any intervals apart. Rotate the line about A in a direction contrary to the motion of the hands of a clock through 30° ; i.e., draw a new line through A, making an angle of 30° with the original line, and locate B and C on it at same intervals as before. Rotate the line about B from its new position, in the same direction, through 20°. Rotate the line about C from its new position, in the same direction, through 15°. What angle does the line in its final position make with its original position ? 20. The same problem as the preceding, there being, however, four angles, 45°, 60°, 30° and 90°. The point in the last two questions is that if a line rotates through various angles and about different points in it, the aggregate rotation is the same as if it all took place about a single fixed point in the line. CHAPTER VI. Parallel Lines. E 1. Parallel lines were defined to be such as have the same direction. Thus the lines in the figure, though differing in position, have all the same direction, and are parallel 2. AC and DE are straight lines. Using the bevel, what do you observe with reference to the magnitudes of the vertically opposite angles ABD and CBE? What with reference to the magnitudes of the angles ABE, DBC ? Draw other intersecting straight lines and note the magnitudes of vertically opposite angles. We may demonstrate the relation between such angles as follows: LABD+ LABE = 2 rt. angles = ZEBC+ LEBA, and dropping from both sides the angle ABE, we have LABD= EBC. Hence if two straight lines cut one another, the vertically opposite angles are equal. Yet such a proposition scarcely needs demonstration; for, as was said in Chapter I., a straight line has the same direction throughout its entire length. Hence the two lines ABC, DBE must deviate from one another as much to the left of B as to the right of B, and thus the angles ABD, EBC are equal. E same B 3. Straight lines which deviate from the same straight line by B the amount, i.e., which make equal angles with this straight line in the same direc- IF tion, must have the same direction, and therefore must be parallel. Thus if the directions, or lines, AB and CD deviate equally from the same direction, or line, EF, i.e., if the angles EAB and ACD are equal, then AB and CD have the same direction, and are said to be parallel. ACD is said to be the interior and opposite angle with respect to the exterior angle EAB. It is to be noted that the parallel lines are inclined to the cutting line equally and in the same direc А с tion. Thus though AB and CD deviate equally from EF, they deviate in opposite directions, and therefore are not parallel. 4. It is understood, then, that if AB and CD are any two parallel lines, and any line GH cuts them, the exterior angle GEB is equal to the interior and opposite angle EFD. 5. The angles AEF, EFD are called alternate angles. By actual measurement, with the bevel or protractor, show that they are equal. We may also prove their equality thus: LGEB= _EFD, because the lines are parallel ; also ZGEB= LAEF, because these are vertically opposite angles; :. LAEF = _EFD. B C : |