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15. When a straight line HO, standing on anotherAB, makes the angle HOA eqnal to the angle HOB; each of these angles is called a right angle; and the line HO is said to be a perpendicular to AB. The measure of the angle HOA is 90 degrees,

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or the fourth part of 360 degrees. Hence a right angle is 90 degrees.

16, An acute angle is less than a right angle; as AOG, or GOH.

17. An obtuse angle is greater than a right angle; as GOB.

18. A plane Triangle is the space enclosed by three straight lines, and has three angles; as A.

19. A right angled Triangle is that which has one of its angles right; as ABC. The side BC, opposite the right angle is called the hypothenuse; the side AC is called the perpendicular; and the side AB is called the base,

A

B

20. An obtuse angled Triangle has one

of its angles obtuse; as A.

A

21. An acute angled Triangle has all its three angles

acute, as in the figure A.

22. An equilateral Triangle has its three sides equal, and also its three angles; as C.

C

23. An isosceles Triangle is that which has two of its sides equal, and the third side either greater or less than either of the equal sides; as D.

24. A scalene Triangle is that which has all its sides unequal; as E.

E

D

25. A quadrilateral figure is a space included by four straight lines. If its four angles be right, it is called a rectangular parallelogram.

26. A Parallelogram is a plane figure bounded by four straight lines, the opposite ones being parallel; that is, if produced ever so far, would never meet.

27. A Square is a four-sided figure, having all its sides equal, and all its angles right angles; as H.

Η

28. An Oblong, or rectangle, is a right angled parallelogram, whose length exceeds its breadth; as I.

I

29. A Rhombus is a parallelogram having all its sides equal, but its angles not right angles; as K.

K

30. A Rhomboid is a parallelogram having its opposite sides equal, but its angles are not right angles, and its length exceeds its breadth; as M.

31. A Trapezium is a figure included by four straight lines, no two of which are parallel to each other; as N.

M

N

A line connecting any two of its angles is called a diagonal.

32. A Trapezoid is a four-sided figure having two of its opposite sides parallel, but the remaining two not parallel; as F.

F

33. Multilateral Figures, or Polygons, are those which have more than four sides. They receive particular names from the number of their sides. Thus, a Pentagon has five sides; a Hexagon, has six sides; a Heptagon, seven; an Octagon, eight; a Nonagon, nine; a Decagon, ten; an Undecagon, eleven; and a Duodecagon, has twelve sides.

If all the sides of each figure be equal, it is called a regular polygon; but if unequal, an irregular polygon.

B

34. The Diameter of a circle is a straight line passing through the centre, and terminated by the circumference; thus A B is the diameter of the circle. The diameter divides the circle into equal parts, each of which is called a semi-circle; the diameter also divides the circumerence into two equal parts, each A containing 180 degrees. Any line drawn from the centre to the circumference is called the radius, as A O, O B, or OS. If OS be drawn from the centre perpendicular to A B, it divides the semicircle into two equal parts, AOS and BOS, each of which is called a quadrant, or one-fourth of the circle; and the arcs AS and BS contain each 90 degrees, and they are said to be the measure of the angles A OS and BOS.

S

35. A Sector of a circle is that part of the circle comprehended under two Radii, not forming one line, and the

part of the circumference between them. From this definition it appears that a sector may be either greater or less than a semi-circle; thus AO B is a sector, and is less than a semi-circle ; and the remaining part of the circle is a sector also, but is greater than a semi-circle.

Α.

T

B

S

36. A Chord of an arc is a straight line joining its extremities, and is less than the diameter; TS is the chord of the arc T HS, or of the arc TABS.

37. A Segment of a circle is that part of the circle contained between the chord and the circumference, and may be either greater or less than a semi-circle; thus TSHT and TABST are segments, the latter being greater than a semicircle, and the former less.

38. Concentric circles are those having the same centre, and the space included between their circumferences is called a ring; as FE.

F

E

PROBLEM I.

To bisect a given straight line A B; that is, to divide it into two equal parts.

From the centres A and B, with any radius, greater than half the given line A B, describe two arcs A intersecting each other at O and S, then the line joining O S will bisect A B.

B

PROBLEM II.

Through a given point x to draw a straight line CD parallel to a given straight line A B.

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sy. Lay the extent or taken on the compasses from s to y; through a y draw C D, which will be parallel to A B.

PROBLEM III.

To draw a straight line CD parallel to A B, and at a given distance F, from it.

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touching these arcs at r and s, and it will be at the given distance from A B, and parallel to it.

PROBLEM IV.

To divide a straight line A B into any number of equal parts.

H

K

G

E

F

D

B

I

Draw A K making any angle with A B; and through B draw BT parallel to A K; take any part A E and repeat it as often as there are parts to be in AB, and from the point B on the line BT, take BI, IS, SV, and V T equal to the parts taken on the line A K; then join AT, EV, GS, HI, and K B, will divide the line AB

which

V

T

into the number of equal parts required, as AC, CD, DF, FB.

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