11. Radii of a Circle. Since the distance between the two points of the compasses is not changed during the construction of a circle, all radii of the same circle are equal. It follows that, all points on a circle are equidistant from the center.

12. Laying off Lengths on Lines. Set the points of the compasses one-half inch apart and lay off one- Ehalf inch on the line EF.

H

a

-F

-K

Set the points of the compasses on the ends of the line a, then lay off a portion of HK equal to the line a. 13. Bisecting a Straight Line. The line AB is to be bisected. With A and B as centers, and with a radius greater than half of AB, describe arcs which intersect each other

Also

Draw lines one, two, three, and four inches long, respectively. draw lines two, four, five, eight, and ten centimeters long, respectively. Bisect these lines by the above method. Test with the ruler or compasses to see if the work is accurate.

Proof of this construction may be found in § 314.

H

B

14. Bisecting an Angle. The angle BAC is to be bisected. With A as a center and any convenient radius, describe arcs intersecting the sides of the angle at H and K. With H as center and with a radius sufficiently great, describe an arc; with K as center and the same radius, describe an arc intersecting the first arc at E. AE. The angle BAC is bisected by the line AE, making angle x equal to angle z.

A

K

Draw the line

With the protractor, draw angles of 50°, 70°, 90°, 120°, 130°, and 150°. Bisect these angles by the above method. Test with the protractor to see if the work is accurate.

Proof of this construction may be found in § 319.

CHAPTER I

STRAIGHT-LINE FIGURES

15. Optical Illusions. Is AB equal to CD, longer than CD, or shorter than CD? Estimate first by sight, only. Test with compasses or ruler.

Is HK equal to EF, longer than EF, or shorter than EF? Can the eye be trusted in comparing the size of objects?

16. Triangle. If a portion of surface is bounded by three straight lines, a triangle is formed. The bounding lines are called the sides of the triangle. The three angles formed by the sides are called the angles of the triangle. The vertices of the three angles are the vertices of the triangle.

Thus ABC is a triangle. The lines AB, BC, and CA are the sides of the triangle. The angles A, B, and C are the angles of the triangle. The points A, B, and C are the vertices of the triangle.

B

B

17. Square. If a portion of surface is bounded by four straight lines, so that all the sides are equal A and all the angles are equal, a square is formed.

Thus ABCD is a square, since the sides AB, BC, CD, and DA are all equal, and the angles A, B, C, and D are all equal.

90°

18. Experiment. Cut out of heavy paper a triangle ABC with the same dimensions as Fig. 1, making AC=4 cm, ZA=40°, ≤C=60°. Place this triangle on Fig. 2 so that AC coincides with DF. The triangles ABC and DEF have a side of one equal to a side of the other. Are these two triangles equal?

Place the triangle ABC on Fig. 3 so that AC coincides with GK. Triangles ABC and GHK have a side and an angle of one equal respectively to a side and an angle of the other. Are these two triangles equal?

Place the triangle ABC on Fig. 4 so that AC coincides with MP. Triangles ABC and MNP have two angles and the included side of one equal respectively to two angles and the included side of the other. Are these two triangles equal?

The following expresses the result of this experiment: 19. If two triangles have two angles and the included side of one equal respectively to two angles and the included side of the other, the triangles are equal.

Proof of this statement may be found in § 248,

20. By means of the ruler, compare the length of the line AB with the lengths of the lines DE, GH, and MN, § 18. In the same way, compare the line BC with the lines EF, HK, and NP. By means of the protractor, compare the number of degrees in ZB with the number of degrees in E, H, and N.

Since AABC=AMNP, it must also be true that AB=MN, BC=NP, and ZB=ZN. Therefore it can be said that:

21. In equal triangles, the corresponding sides are equal and the corresponding angles are equal.

1. Place the triangle which was cut out equal to Fig. 1, § 18, so that AC coincides with EF of Fig. 5. Then place the same triangle so that AC coincides with GH of Fig. 6. Is it equal to each of these triangles? Does the position of a triangle affect its size?

2. The figure AFGC is a square. B is the middle point of AC. The lines DB and EB are drawn making the angles ABD and CBE each equal to 60°. Prove that ABDA = ABEC.

Proof.

ZA=ZC.

(Since the angles of a square are equal.)

AB=BC

$ 17

F

D

Given

(By the statement that B is the middle point of AC.)

G

E

(If two triangles have two angles and the included side of one equal respectively to two angles and the included side of the other, the triangles are equal.)