A straight line may be found equal to the sum of two lines by placing them in the same straight line with a common extremity, so that a point which by its motion has described the one, would, if its motion were continued in the same direction, immediately proceed to describe the other. The operation of adding is expressed by the sign +. Thus AB+BC= AC. A B Similarly, the difference of two lines may be found by placing them in the same straight line, with a common extremity, so that a point which has described the one, would, if its motion were reversed, immediately proceed to describe the other. The operation of subtracting is expressed by the sign –. The sign placed between two quantities means that the less is to be subtracted from the greater; thus AB-AC= BC. Lines are expressed arithmetically as multiples of some known length. Thus we speak of 7 yards, 11 miles, a yard and a mile being known lengths. EXERCISES ON STRAIGHT LINES. I. Prove that the sum and difference of two lines are together double of the greater of them. 2. AB is divided into two equal parts in O, and P is any point in the line AB: prove that (AP~ BP) = OP. is any point in AB (AP+BP) = OP. 3. On the same supposition if P produced through A or B, prove that Angles. Def. 6. Two straight lines that meet one another form an angle at the point where they meet: and the lines are called the arms of the angle; and the point the vertex of the angle. An angle is a magnitude, and is measured by the quantity of turning that one of its arms must undergo in order to be brought to coincide with the other. It is obvious that the magnitude of an angle does not depend on the length of its arms. A B An angle may be conceived as generated by the rotation of an arm round its extremity, the motion being in one plane. An angle is named by a single letter at the vertex, as A ; or by a letter at the vertex placed between letters on each of the arms. any Ax. 4. Ax. 5. Angles are equal when they could be placed on one another so that their vertices would coincide in position, and their arms in direction. Thus the angles B, E are equal, if when E is placed on B, and EF on BC, then ED has the same direction as BA. Conversely, angles which are equal can be conceived as placed on one another so as to coincide. And angles which are unequal could not coincide. The sum of two angles is found by placing them so as to have a common vertex and a common arm, and on opposite sides of the common arm, so that an arm which by its rotation has described one angle, would, if its motion were continued in the same direction, immediately proceed to describe the other. Angles so placed are said to be adjacent. The difference of two angles is found by placing them on the same side of the common arm, so that the arm which has described the first angle would, if its motion were reversed, immediately proceed to describe the other. Thus AOC is the sum of the angles AOB, BOC; and AOB is the difference of AOC and BOC. C B A Def. 7. The bisector of an angle is the line that divides it into two equal angles. It is obvious that an angle can have one, and only one bisector. Thus if the angle AOB = the angle BOC, then OB is the bisector of the angle AOC. EXERCISES ON ADDITION AND SUBTRACTION OF ANGLES. I. Prove that if an angle AOC is bisected by OB, and divided into two unequal angles by OP, then AOP-COP=2 BOP. 2. Prove that if an angle AOC is bisected by OB, and a line OP is drawn outside the angle AOC, then AOP+COP=2 BOP. 3. The sum and difference of two angles are together equal to twice the greater angle. Def. 8. A line may be conceived to revolve continuously in one direction round its extremity until it returns once more to its initial position. It is then said to have made one revolution. B A Def. 9. When it coincides with what was initially its continuation, it has described half a revolution, and the angle it has then described is called a straight angle, because the arms of it form one straight line. Thus the angle AOB is a straight angle. All straight angles are equal to one another. Ax. 2 and 5. Def. 10. Half a straight angle, or a quarter of one revolution, is called a right angle. Thus if AOC and COB are equal, each of them is a right angle, or half a straight angle, or a quarter of a revolution. All right angles are therefore equal to one another. Def. II. A straight line is said to be perpendicular to another straight line when it makes a right angle with it. Hence there can be only one perpendicular to a given line at a given point, on one side of that line, because only one line can make a right angle with the given line at that point. Def. 12. An acute angle is one which is less than a right angle. Def. 13. An obtuse angle is one which is greater than a right angle. Def. 14. A reflex angle is one which is greater than a straight angle. THEOREM I. When a straight line stands upon another straight line, it will make the adjacent angles together equal to two right angles: and conversely, if two adjacent angles are equal to two right angles, the exterior arms of these angles will be in the same straight line. Let DB stand upon the straight line AC; then will the adjacent angles ABD, DBC be together equal to two right angles. A D B C Proof. For the sum of ABD and DBC is the straight angle ABC, which is equal to two right angles. Def. 10. Conversely, if ABD and DBC are together equal to two right angles, AB and BC will be in one straight line. Proof. For ABD and DBC make up a straight angle by supposition, and therefore the arms of it, BA and BC, will be in one straight line. Def. 9. Def. 15. Two angles are said to be supplementary to one another when their sum is a straight angle. Def. 16. Two angles are said to be complementary to one another when their sum is a right angle. |