An acute angle is less than a right angle. Ex. ABC

B

14. A circle is a figure contained by one curved line, which is called its circumference, and is such that every portion of it is equidistant from a certain point within it called its centre. Ex. A

.A

15. An arc is any portion of the circumference. Ex. AB

B

A

16. A radius (plural radiï) is a straight line drawn from the centre to the circumference. Ex. AB

B

17. A diameter is a straight line drawn through the centre, and terminated at both extremities by the circumference. Ex. AB—

B

B

18. A semicircle is half a circle, and it is contained by a diameter and half the circumference. Ex. ABC

B

A

19. A tangent is a straight line which meets a circle, and, being produced, does not cut it. Ex. AB. The point where it touches the circle is called the point of contact. Ex. B.

B

NOTE 1.-The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, marked. Hence, a semicircle will contain 180°; a quarter of a circle, or a quadrant, 90°; the sixth part, 60°, &c.

NOTE 2.-Angles at the centre of a circle are proportional to the arcs on which they stand. Hence, a quadrant will contain an angle of 90°, i.e., a right angle. A degree is divided into 60 equal parts, called minutes, marked'; and each minute into 60 equal parts, called seconds, marked ". Thus, 44° 30′ 27′′ reads-44 degrees, 30 minutes, 27 seconds.

Problem 1.

To bisect a given straight line AB, or a given arc AB; that is, to divide it into two equal parts.

1. From point A as centre, with any radius greater than half the line AB, describe the arc DE.

2. From point B as centre, with the same radius, describe the arc FG, intersecting the arc DE in H and K.

3. Draw the straight line HK, and the given straight line AB will be bisected in the point L.

NOTE 1.-HK is perpendicular to AB, and at right angles to it. NOTE 2.-The same method is to be followed in bisecting the given arc AB.

Problem 2.-(A.)

To draw a straight line perpendicular to a given straight line AB, from a given point in the line.

First. Let the given point C be at or near the middle of the line AB.

1. From point C as centre, with any convenient radius, describe a semicircle meeting AB in points D and E.

2. From point D as centre, with any radius, describe arc FG; and from E as centre, with the same radius, intersect the arc FG in the point H.

3, Draw the straight line HC, and it will be perpendicular the given straight line AB.

NOTE 1.-Because HC is perpendicular to AB, each of the angles ACH, BCH is a right angle.

NOTE 2.—In naming an angle, the middle letter should be at the angle.

(B.)

Secondly. Let the given point B be at or near one end of the line AB.

A

C

ΕΠ

1. From point B as centre, with any convenient radius describe an arc CDE.

2. From point C, with the same radius, cut the arc in D; from D, with the same radius, describe an arc EF, cutting CDE in E; and from E, with the same radius, cut the arc EF in F.

3. Draw the line FB, and it will be perpendicular to, or at right angles to, the given straight line AB.

Problem 3.-(A.)

To draw a straight line perpendicular to a given straight line AB, from a given point outside it.

First. Let the given point C be opposite, or nearly opposite, the middle of the line AB.

1. From point C, with any sufficient radius, describe an arc cutting AB in D and E.

2. From points D and E as centres, with any radius, describe arcs cutting each other in the point F.

3. Draw the line CF, and it will be perpendicular to the given straight line AB.

(B.)

Secondly. Let the given point C be opposite, or nearly opposite, one end of the line AB.