a folded sheet of paper as a straight edge draw a line that looks three inches long. Measure it with your ruler. When you measure a line be sure to place the ruler so that the very end of an inch or centimeter space falls exactly on the end of the line to be measured. If your ruler is old and the end is worn, you will measure more accurately to start, not at the end, but at the point marked 1.

How long is the line that you drew? How much was your error? Try it again. Did you do better this time? First estimate and then measure the length and width of your book by both linear and metric measures. Do the same for the top of your desk. The picture on the opposite page shows Mary Kiernan measuring the length of a crayon box. How long is the crayon box in your room?

Measuring Longer Lines.- When we want to measure a long line like the length of a street or the distance between two towns we get surveyors or civil engineers to measure it. Surveyors use in place of a ruler a steel tape 100 feet long, divided into feet and tenths of a foot. An instrument called a transit is used to keep them in a straight line.

Broken and Curved Lines.-In the chapter on graphs we had pictures of broken lines but we did not define them. A line made up of several line segments running in different directions is called a broken line.

A curved line or curve is a line no part of which is straight.

The Circle. The most important curved line is called a circle. It is a closed curve which is everywhere the same distance from a point inside called the center. Can you draw a good circle free-hand? Try it. Now

try it again, using a circular object such as a coin. Name other circular objects with which you might draw circles. Could you draw a circle of any size that you might want that way?

Compasses. Dick Clark had so much trouble drawing circles of the sizes he wanted that his father gave him a pair of compasses. This consists of two legs of the same length, with a movable joint between them. One leg ends in a sharp mental point, the other in a pencil or pen point.

USING COMPASSES

The distance from any point on a circle to its center is called the radius of the circle. To draw a circle with a radius of one inch we open the compasses so that the distance between the end points is one inch. (Use a ruler to determine this distance.) Then we thrust the sharp point of the compasses into the paper and turn the pencil point completely around, forming a closed curve. This is a circle. The point where the sharp end pierced the paper is the center of the circle.

Radius

Center

A

B

Arc

Any part of the curve of the circle is called an arc. Many arches are arcs of circles. All lines made by

the compasses are arcs, when they stop short of a complete circle.

A chord is a straight line joining two points on a circle; as the chord AB. If a chord passes

A

B

D

through the center, it is a diameter; as the line CD. A diameter consists of two radii extending in opposite directions from the center.

Name some common circular objects.

Angles. - As the hands of a clock rotate, they form various two-sided figures, called angles. In the same way, if a radius turns (rotates) about the center of its circle, it makes different angles with other radii of the circle. In the figure, angle ABC shows how much the radius AB must turn about point B to reach the position of CB. The radii AB and CB are called the sides of the angle and point B is its vertex.

As we cannot conveniently picture an angle as the rotation of a radius, we may represent it by drawing two radii of a circle. The angle is measured by the amount of rotation or opening between the two

B

C

-A

radii. Since all the radii of a circle are drawn from the same point, we may say, then, that an angle is the figure formed when two straight lines are drawn from the same point. In naming an angle we usually place a letter at the vertex, and one at the other end of each side, as shown in the diagram. When only one angle is named at a point, we may identify it simply by the vertex letter. Thus this angle may be called "the angle B."

corner. How many right angles can you see from where you sit?

When two lines form a right angle each line is said to be perpendicular to the other. Thus AB is perpendicular to CD.

B

A straight angle is an angle whose sides lie in the same straight line but extend in opposite directions. Thus, two radii which form a diameter make a straight angle.

C

+

A

A

C

Obtuse

Angle

Two lines are oblique to each other when they make

-D

B

Two angles are complementary if their sum equals a right angle, as

angles BCD and ACD or angles 1 and 2.

B

C

their sum equals a straight angle,

as angles ECF and BCF or angles 3 and 4.

Two complementary angles are called complements of each other, and two supplementary angles, supplements.