Draw F G parallel to A C.

Draw G H and F I parallel to D B.

Join H I, which will complete the square in the trapezium.

To inscribe a Circle in a given Trapezium, A B C D, of which the adjacent sides are equal. (Fig. 38.)

C

Fig. 38.

Draw the diagonal A B, which will bisect the angle CB D, or C A D.

Bisect the angle A D B.

Produce the bisecting line until it cuts A B in O.

Then O is the centre from which a circle may be described, touching all four sides of the trapezium.

To trisect* a Right Angle, AB C. (Fig. 39.)

Fig. 39.

From B, with any radius, describe the quadrant D E.

*Trisect. To cut into three equal parts.

From D, with the radius D B, describe an arc cutting E D in F.

From E, with the same radius, describe an arc cutting ED in G.

Draw lines B F and B G, which will trisect the right angle.

The Measurement of Angles. (Fig. 40.)

Angles are estimated according to the position which the two lines of which they are formed occupy as radii of a circle.

The circle being divided into 360 equal parts, called "degrees," it will be evident that the lines A, O, C, contain 90 degrees (written 90°) or a right angle.

Similarly B OC is a right angle.

Now, if these right angles be trisected (as per last problem), each of the divisions will contain 30°, thus :

A O B is in reality not any angle at all, being a perfectly straight line; but the slightest divergence from it would cause it to become an angle; as 179°, &c.

Each of these angles being again divided into three parts will give tens, which may again be divided into units; and thus angles may be constructed or measured with the greatest accuracy.

Example No. 1 of the foregoing. (Fig. 41.)

To find the angle contained by the lines A B C.

Erect a perpendicular at B.

Draw the quadrant D E, and trisect it.

Divide the arc GE into three equal parts by points H and I. (70° and 80°.)

Bisect the arc H I, and it will be seen that the line B C falls precisely on the bisecting point.

A B C is therefore an angle of 75°.

Had the line B C not fallen exactly in the bisecting point, further subdivision would have been necessary.

To construct at a given point B an angle of a required number of degrees, say 100°.

At B erect a perpendicular, B C.

Trisect the right angle, carrying on the arc_beyond the perpendicular, C.

Divide any one of the three divisions into three equal parts representing tens.

Set off one of these tens beyond C, viz., to D.

Draw B D.

Then A B D will be an angle of 100°.

To construct a Triangle, when the length of the base and the angles at the base are given. (Fig. 43.)

Let it be required that the base should be 2'5 2 (deci

mal 5, or 2 and 5 tenths, which is 2) inches long, that the angle at A should be 50°, and that at B 45°.

Draw the base 2.5 inches long.

At A erect a perpendicular; draw a quadrant and trisect it in E D.

Divide the middle portion, D E, into three equal parts, and the second division from E will be 50°.

Draw a line from A through point 50 and produce it. At B erect a perpendicular, and bisect the right angle thus formed (as 45° is one-half of 90°).

Produce the bisecting line until it meets the line of the opposite angle in F.

Then A B F will be the required triangle.

NOTE.-All the three angles of a triangle are always equal to two right angles, that is 180°, and therefore, as one of the above angles is 50°, and the other 45°-total 95°-the vertical angle, that is, that opposite the base, will be 85°.

The Protractor. (Fig. 44.)

For measuring and constructing angles, there is, in most cases of mathematical instruments, a brass semicircle called a Protractor. This has a short line marked at C, and two rows of figures round the rim- the one reading from right to left, and the other the reverse way.

In order to measure an angle by means of the protractor, place the edge A B on the straight line which is to form one of the sides of the angle, with the point C exactly against the point of the angle to be measured. Then the line C D will be seen to correspond with the point 60°, and B C D is therefore an angle of 60°; or, reading from the left side, A C D is an angle of 120°.

In constructing an angle, place C against the point at which it is desired to construct an angle; mark a point on your paper exactly against the figure corresponding to the number of degrees required; remove the protractor, and draw a line through the point thus obtained, to C, which will give the desired angle.

Protractors are sometimes made of wood or ivory, and of a rectangular form, as E F. These are used in a manner precisely similar to the semicircular instruments, but are not generally thought as useful or exact in practice.