8. Two lines which intersect in the centre of a circle include one. eighth of the circumference: give in degrees the magnitudes of the angles formed. 9. A straight line makes with another straight line two angles, one of which is four times the other; find the magnitude of the angles in degrees. 10. A number of straight lines meet in a point and include equal angles, each angle being 13° 20′; how many angles are there ? II. There are seven equal angies formed at the centre of a circle: what is the magnitude of each of the intercepted arcs? 12. What is the supplement of 28° 15′ and of 113° 15′ 22′′ ? 13. What are the complements of 15° 10′ and 17° 41′ 52′′ ? 14. Seven roads meet at a point, the angles being as follows:Between 1st and 2nd Between 4th and 5th ... 33° 102° ... ... 45° ... 25° 78° ... 32° Find the angle between the 1st and 7th, and say whether any two roads are in the same straight line. Easy Exercises. (a). Draw a line, MN, two inches long, and, with its extremities as centres and a radius of one inch and a half, describe two equal circles. Join carefully the points of intersection of these circles; and, by means of the compasses, show that the line thus drawn bisects M N. (b). Draw a line two inches long, and, with its extremities as centres and a radius of one inch, describe two circles. (c). Draw any straight line, A B, and, with A and B as centres, describe two equal circles large enough to intersect. Join the points of intersection, and show by measurement that A B is bisected by this line. (d). Make a right angle, and name it ABC; cut off any length, BA, on one side, and any length, BC, on the other. Join the points A and C. Next find the middle point of A C, and call it M. Join M B. Finally measure A C. Also measure M B, and say what is the relation between their lengths. (e). Express in degrees, minutes, and seconds the angle between the hands of a watch at 10, at 11.10, at 6.30, at 1.25, and at 9.50. Mark on paper the positions of two points whose distance shall be less than an inch; and then find a point which shall be one inch from each of them. THEOREMS ON CHAPTER III. THE CIRCLE. DEFINITIONS. 25. A circle is a plane figure, bounded by one line, called the circumference, such that all straight lines drawn from a certain point called the centre to the circumference are equal. 26. Any straight line drawn from the centre to the circumference is termed a radius. 27. Any straight line passing through the centre, and terminated by the circumference, is called a diameter. 28. Any straight line within the circle, and terminated by the circumference, is termed a chord. 29. A portion of the circumference is termed an arc. 30. An angle whose vertex is at the centre is termed an angle at the centre, and is said to stand on the arc intercepted by its sides. THEOREM IV. In equal circles equal angles at the centre stand on equal arcs : And equal arcs subtend equal angles at the centre. Imagine the circle A B C placed on the circle E F G, so that the centre D may fall on the centre H, and the line DB on the line H F, then because BDC== === 4FHG, DC will fall on HG; and because the circles are equal, the circumference A B C will coincide with the circumference E F G. Therefore the point B will fall on F, and C will fall on G. Hence the arc B C coincides with the arc FG, and is equal to it. Again, if the arc BC be equal to the arc FG, then shall the angle B D C be equal to the angle F HG. Suppose the circle A B C placed in E F G, so that D may fall on H and D B on H F, then, because the circles are equal, the circumferences will coincide; therefore the point B will fall on F. Because the arc B C is equal to the arc F G, the point C will fall on G, and consequently DC will fall on H G. Therefore the angle BDC coincides with the angle FH G, and is equal to it. CHAPTER IV. TRIANGLES. SECTION I.-The Equality of Tringles. Definition.--Mark upon a sheet of paper the position of a point, A. At the point A, form an angle by drawing from A two straight lines, AB, A C, in different directions. Take a point, B on AB, and a point, C on A C, and draw the line BC; then B C and A C will form an angle at C, and CB and AB will form an angle at B. The three straight lines form a closed figure, having three angles, and therefore termed a triangle. B FIG. 46. Meaning of the Terms Hypothesis and Conclusion.-Upon a sheet of paper draw two straight lines from the same point, so as to form an angle, M; also draw two straight lines, b and c, of any length. Cut out the angle M, and, by placing it upon another sheet of paper and tracing the sides with a HA AA Mark off from these sides join the points B and C. pencil, make an angle, B A C, equal to M. A C, equal to b, and A B, equal to c, and A triangle will thus be made, having an angle equal to M and the sides which contain this angle respectively equal to the lines b and c. Now, in the same way, make another angle, E D F, equal to M, and from the sides mark off DE, equal to A B, and D F, equal to A C. In drawing these triangles, we have paid no regard to the lines B C and EF; but it will be found that these lines are equal. Again, we made the angles at A and D equal to the angle M, and, therefore, equal to one another; but we paid no regard to the angles at B and E, or to the angles at C and F; nevertheless, it may be shown that the angle B is equal to the angle E, and the angle C to the angle F. The statement or enunciation of the proposition by which these equalities will be proved may be made as follows: 1. When two triangles have two sides of the one respectively equal to two sides of the other, and the included angles equal; 2. The third sides are equal, and the remaining angles of the one are respectively equal to the remaining angles of the other. There are two parts to this enunciation. The first part contains a statement of something which is known or granted respecting the triangles, and this statement is termed the hypothesis. The second part of the enunciation is the statement of some fact which will be found necessarily to follow from the hypothesis. This is termed the conclusion. Angle B angle D, Angle A angle F, and also These facts, therefore, constitute the hypothesis. We then find that angle E, and angle C Side B C = = = side EF. These facts constitute the conclusion. The enunciation of every proposition consists of hypothesis and conclusion. As another illustration, let us draw a straight line, a, and cut out A A a two angles, M and N. On another sheet of paper let us draw a straight line, and cut off a length, BC, equal to a. By placing the angle M in contact with the straight line B C, and drawing a pencil's point from B along the free side, let us make an angle, CBA, equal to M. Similarly, by placing the angle N FIG. 49 on the straight line BC so that both sides terminate in the point C, let us make an angle, BCA, equal to N. By this means we shall have made a triangle, ABC, having a base, BC, equal to a, an angle, ABC, at one extremity of the base equal to M, and an angle, B CA, at the other extremity of the base equal to N. A A B FIG. 50. Again, repeat the process on another base, EF, equal to a, making the angle FED equal to M and the angle E F D equal to N. The two triangles suggest the following proposition : |