but we must learn to construct an angle equal to any given angle. This is how we do it.

Let ABC be the given angle. We take the line HK for one side of our new angle, and decide to use H for its vertex.

Place the sharp point of the compasses on B and draw an arc cutting the sides of the angle ABC at M and N. With the same opening, place the point on H and draw a similar arc cutting HK at D. Set the compasses at N and open them so that the pencil point is at M. Using this opening, place the point at D and draw an arc cutting the other arc at E. Draw EH. Then ZEHK is equal to the ZABC. Prove this with your protractor.

Constructing and Testing Angles

1. With your protractor draw an angle of 35°. Now draw a straight line and without the protractor but with the compasses construct an angle of the same size at some point on the line.

2. Draw an obtuse angle. Construct with the compasses an angle equal to it.

3. Test the accuracy of your work by measuring the angles with your protractor.

4. Draw a line segment, and mark down a point outside of it. Construct a perpendicular line from that point to that line. With your protractor measure the two angles formed. Are they right angles? Are they equal?

Fixing the Habit of Systematic Review

Try to form the habit of systematically reviewing back work which you know is a stepping stone to new work. "Watch your step," both the last step and the next step. Remember that system will save time and trouble for you. "Do it now "is the only sure way to form the good habit of systematic review.

1. With your protractor construct the following angles : 30°, 120°, 45°, 150°, 50°.

2. What is the complement of 30°? of 45°? of 50°? of 66°? of 89°?

3. What is the supplement of 120°? of 150°?

4. Bisect each of the angles in example 1 without a protractor.

5. I. R. B. [1] Construct a broken-line graph of the following yearly incomes. Choose a convenient unit on the horizontal line for the time and on the vertical line for the money. Copy your graph in your I. R. B.

High School
Training

In School In School $500 $750 $1000 $1150 $1550

No High
School
Training

$200

$250 $350 $470 $575 $600 $688

Does a high school education pay?

If you want to get ahead and stay ahead, you must use your head and stay on the school job.

Triangles. - On the school picnic the girls made a table by setting a rickety box on its side. The box had no top or bottom, so it settled and finally broke down. George Burns set it up again and nailed a stick slantwise across the open top so as to make a triangle. This time it held up. Can you tell why? The beauty of a triangle is that it is stiff and rigid. Steel bridges have to be rigid; they always show many triangles. It is this same rigid or fixed quality which makes triangles so valuable in mathematics, especially for fixing points. In much of the work we have just been doing we have really been using triangles, even if we did not always draw the sides.

MAKING THE BOX RIGID

1

A triangle is a plane 1 figure bounded by three straight lines, like the figure ABC.

α

The three lines are called the sides and the points where they meet are called vertices (each one is a vertex). Thus a triangle has three sides, three vertices, and three angles.

B

Kinds of Triangles. When the sides and angles differ in size, they make different kinds of triangles. An acute triangle is a triangle in which all three angles are acute.

An obtuse triangle is a triangle in which one angle is obtuse.

A right triangle is a triangle in which one angle is a right angle.

RIGHT A

OBTUSE A

ACUTE Δ

The hypotenuse is the longest side of a right triangle. Which angle is it opposite? How many acute angles are there in a right triangle? in an obtuse triangle?

An equilateral triangle is a triangle which has all of its sides equal. Equilateral triangles are also equiangular, that is, all three angles are equal.

An isosceles triangle is a triangle which has two sides equal.

A scalene triangle is a triangle whose sides are all unequal.

1 By a plane we mean something smooth and flat like the top of a desk or smooth pond. Can you mention some other examples of planes?

to draw lines of a definite length and how to make angles of a definite size. Now we are going to put these things together to make different kinds of triangles.

(a) Constructing a Triangle with Three Given Sides. - If we wish to make a triangle whose sides are equal respectively to three line segments a, b, and c, we first draw a line AX of

any convenient length. From A we mark off with the compasses a distance AB equal to the longest segment c. Then we open the compasses a distance equal to the segment b and placing the sharp point at A we swing an arc as shown in the diagram. Finally with B as center and a radius equal to the segment a we draw an arc crossing the first arc at the point C. Then we draw the straight lines AC and BC. The triangle ABC has sides equal to the given lines.

With your ruler measure the line segments a, b, and c, and then measure the sides of the triangle and check the work.