Suppose we draw four lines on a piece of paper, ab and cd. These intersect at a point, m. We have then four spaces marked out, and called angles. The four angles are in the diagram all the same size, and are termed right angles, and the lines containing them are perpendicular to each other.
Fig. 503.—Right angles.
But by altering the position of the lines (see fig. 504), we have two pairs of angles quite different from right angles; one angle, a´ m´ c´, is smaller, while a´ m´ d´ is much larger than the right angle. The former kind are called acute, the latter obtuse angles. We can therefore obtain a great number of acute angles, but only three obtuse, and four right angles around a given point, m.
Fig. 504.—Obtuse and acute angles.
The length of the sides of an angle have no effect on its magnitude, which is determined by the inclination of the lines towards each other. We now may consider the magnitude of angles, and the way to determine them. For this purpose we must describe a circle, which is figured in the diagram. But what is a circle?—A circle is a curved line which always is at the same distance from a certain fixed point, and the ends of this line meet at the point from which the line started.
Fig. 505.—The circle, etc.
If we fasten a nail or hold a pencil on the table, and tie a thread to it, and to the other end of the thread another pencil, we can describe a line around the first pencil by keeping the thread tightly stretched. This line is at all points at equal distance from the centre point. Any line from the centre to the circumference is called a radius, and a line through the centre to each side of the circumference is the diameter, or double the radius. The circumference is three (3·14) times the diameter. Any portion, say k i l, is an arc, and the line, k l, is the chord of that arc. A line like m n is a secant, and o p is a tangent, or a line touching at one point only.