6. What is a surface? A Surface is that which has length and breadth, without height or thickness. 7. What is a plane, or plane surface? A Plane is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide. 8. When are lines said to be parallel? Two straight lines are said to be parallel when they are at the same distance from each other at every point. Parallel lines will never meet each other. 9. When are two curves said to be parallel? Two curves are said to be parallel or concentric, when they are at the same distance from each other. Parallel curves will not meet each other. 10. What are oblique lines? Oblique lines are those which approach each other, and meet if sufficiently prolonged. 11. What are horizontal lines? Lines which are parallel to the horizon, or to the water level, are called Horizontal Lines. Thus, the eaves of a house are horizontal. 12. What are vertical lines? All plumb lines are called Vertical Lines. Thus, trees and plants grow in vertical lines. 13. What is an angle? How is it read? An Angle is the opening or inclination of two lines which meet each other in a point. Thus the lines AC, AB, form an angle at the point A. The lines AC, and AB, are called the sides of the angle, and their intersec- A tion A, the vertex. B The angle is generally read by placing the letter at the vertex in the middle: thus, we say the angle CAB. may, however, say simply, the angle A. 14. May angles be formed by curved lines? Yes, either by two curves, CA, BA A forming the angle A, called a curvilinear angle: Or, by two curves AC, AB, forming the angle A: Or, by a straight line and curve, which is called a mixtilinear angle. We B A B 15. When is one line said to be perpendicular to another? · One line is perpendicular to another, when it inclines no more to the one side than to the other. Thus, the line DB is perpendicular to AC, and the angle DBA is equal to DBC. D A B 16. What are the angles called? When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC are right angles. Hence, all right angles are equal to each other. 17. What is an acute angle? An acute angle is less than a right angle. Thus, DBC is an acute angle. 18. What is an obtuse angle? An obtuse angle is greater than a right angle. Thus, EBC is an obtuse angle. 19. If two lines intersect each other, what follows? If two lines intersect each other, the opposite angles A and A are called vertical angles. These angles are equal to each other, and so also are the opposite angles B and B. E 20. What follows when two parallel lines are cut by a third line? If two parallel lines CD, AB, are cut by a third line IG, the angles IHD and AFG, are called alternate angles. These angles are equal to each other. The angle IHD is also A C G -B -D H equal to the angle IFB, and to the opposite angle CHG. 21. What follows when a line is perpendicular to one of several parallel lines? If a line be perpendicular to one of several parallel lines, it will be perpendicular to all the others. Thus, if AB, CD, and EF, be parallel, the line CH drawn perpendicular to AB, will also be perpendicular to CD and EF. 22. How many lines can be drawn from one point perpen dicular to a line? From the same point D, only one line DB, can be drawn, which will be perpendicular to AB. A D 23. If oblique lines are also drawn, what follows? If oblique lines be drawn, as DC, DF, then :1st. The perpendicular DB, will be shorter than any of the oblique lines. 2d. The oblique lines which are nearest the perpendicular, will be less than those which are more remote. within, called the centre. Thus, if all the points of the curve AEB are equally distant from the centre C, this curve will be the circumference of a circle. B 25. For what is the circumference of a circle used? The circumference of a circle is used for the measure ment of angles. 26. How is it divided? It is divided into 360 equal parts called degrees, each degree is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds, are marked thus, °, ', "; and 9° 18′ 10′′, are read, 9 degrees, 18 minutes, and 10 seconds. 27. How are the angles measured? 90 80 E 40 D Suppose the circumference of a circle to be divided into 360 equal parts, beginning at the point B. If, through 180 the point of division marked 40, we draw CE, then, the angle ECB will be equal to 40 degrees. If we draw B CF through the point of division marked 80, it will make CB an angle equal to 80 degrees. 28. How many degrees are there in one right angle,—in two-in three-in four? If two lines AB, DE, are perpendicular to each other, the four angles BCD, DCA, ACE, and ECB, will be equal. These two lines will divide the circumference of the circle into the four equal parts BD, DA, AE, and EB, and each part will measure one D B of the right angles. But one quarter of 360 degrees, is 90 degrees. Hence, one right angle contains 90 degrees, two right angles 180 degrees, three right angles 270 degrees, and four right angles 360 degrees. 29. What is one quarter of the circumference called?— one half of it? One quarter of the circumference is called a quadrant, and contains 90 degrees. One half of the circumference is called a semi-circumference, and con |