angle. Applying the ruler, it will be found that CA and AB are in the same straight line. Hence it appears that the three angles of any equilateral triangle are together equal to 180°, and any one of the angles in such a triangle is 60°.

Measure the angles in several of the equilateral triangles with the protractor to verify this.

2. Take a line AB of, say, 25 millimetres in length, and with centres A and B describe portions of circles intersecting as indicated. at C, each circle having the same radius, say 35 millimetres. Draw lines from C to A and B. Then the triangle CAB has two sides equal. A triangle with two of its sides equal is called an isosceles A triangle.

B

Adjusting the bevel to the angles CAB and CBA, compare their magnitudes.

Compare also the sizes of these angles by accurately cutting the triangle out of the paper, and placing the triangle reversed in the vacant space left in the paper, so that the angle B rests in the space A.

Compare also the sizes of these angles by folding the triangle along the line from C to the middle of AB. Construct the following isosceles triangles:

Base 1 in., each side 2 in.

Base 3 in., each side 2 in.
Base 23 in., each side 218 in.

In each case compare the magnitudes of the angles at the base.

The result of our observations is that the angles at the base of an isosceles triangle are equal. Of course it would follow from this that all the angles in an equilateral triangle are equal, as we have already

seen.

A

Prolong the sides CA, CB, and adjusting the bevel to the angles BAD, ABE, on the other side of the base, they will be found to be equal. This may also be reasoned out as follows: The angles on one side of a straight line at any point in it make up 180°. But the angles CAB and CBA are equal.

B

E

Therefore the remaining angles BAD and ABE are also equal.

3. Taking any line AB, with the bevel or protractor construct equal angles at A and B, and produce the bounding lines of these angles to meet in C.

Then employing the

dividers or compasses, compare the lengths of the sides CA, CB, of the A triangle CAB.

Construct the following triangles:

Base 25 millimetres, each of angles at base 75°.
Base 70 millimetres, each of angles at base 30°.
Base 3 in., each of angles at base 45°.

B

In each case compare the magnitudes of the sides adjacent to the equal angles.

The result of our observations is that if two angles of a triangle are equal, the sides opposite to these angles are also equal.

In the case of each of the above triangles measure the size of the angle at the top, or vertex, of the triangle, and find the total number of degrees in the three angles of each triangle.

4. Take a line AB of length 35 millimetres, and with centres A and B, and radii 45 and 50 millimetres respectively, describe portions of circles, so that they intersect at C. Join CA and CB. We have thus a triangle CAB whose sides. are unequal, called a scalene triangle.

Construct the following triangles :

A

45

With sides 3, 5 and 6 inches.

With sides 70, 80 and 100 millimetres.
With sides 3, 4 and 2 inches.

With the bevel lay off three angles adjacent to one another, equal to the angles of each triangle, in the way indicated in the adjacent figure; and determine the positions of the initial and final lines, LM, LK, with respect to one another.

K

M

B

What conclusion do you draw as to the total number of degrees in the three angles of each of these triangles?

Can you construct a triangle with sides of 30, 50 and 90 millimetres, or with sides of 2, 3 and 6 inches? Attempt the construction.

What relation must exist between the given lengths, that a triangle may be constructed with sides of such lengths?

Exercises.

Teachers are advised to have their classes work but a few of the exercises at the close of each chapter. The time of pupils should be chiefly occupied in verifying the geometric truths reached in the text.

1. At a given point in a straight line construct an angle of 60°, using only compasses and ruler.

2. Construct an isosceles triangle, and produce the base both ways. What do you note as to the magnitudes of the exterior angles so formed?

3. Construct a triangle with sides 30, 50, 70 millimetres. With the bevel or protractor determine which is the greatest angle and which the least.

4. The angle at the vertex of an isosceles triangle is 75°, and each of the equal sides is 2 inches. Construct the triangle.

5. At A in the line AB construct the angle BAD of 40°, and at B the angle ABC of 120°. Produce AD, BC to meet. Measure the size of the third angle of this triangle. Which is the greatest side and which the least?

6. On one side of BC describe an equilateral triangle ABC, and on the other side of BC describe an isosceles triangle DBC. Join AD. Take a number of points E, F, G, . . . in AD. What do you note as to the lengths of EB and EC; of FB and FC; of GB and GC,

?

7. Make the same construction as in the preceding question, but with the isosceles triangle on the same side of BC as the equilateral. Produce AD both ways. What again do you note as to the distances of any point in AD, or AD produced, from B and C ?

8. On BC describe an equilateral triangle ABC, and on the other side of BC describe a scalene triangle DBC. Join AD. Take a number of points E, F, G, . . . in AD. What do you note as to the lengths of EB and EC; of FB and FC; of GB and GC,

9. Repeating the figure of 6, take in BC, and on the same side of AD, a number of points K, L, M, N. .. What do you note as to the lengths of AK, AL, AM, AN, .? Do they seem to follow any law as to magnitude?

10. Describe an equilateral triangle ABC. On BC describe an equilateral triangle DBC; on CA an equilateral triangle ECA ; and on AB an equilateral triangle FAB. Join AD, BE, CF. What do you observe as to the positions of the lines DC, CE with respect to one another; of EA, AF; and of FB, BD?

11. In the preceding question mark all the angles that are equal to one another; also all the lines that are equal to one another.

What triangles are isosceles?

Do you observe any equilateral four-sided figures?

How many equilateral triangles are there?

12. With centre A, outside a straight line, describe a circle of such radius as to cut the line in two points, B and C. What sort of triangle is ABC?

13. In the figure of the preceding question find on the side of BC remote from A, a point D, such that a circle with D as centre can be described to pass through both B and C.

14. B and C being two points in a line, find on either side of the line points K, L, M, N, . . . such that a circle may be described, with any one of them as centre, to pass through B and C. What do you observe as to the positions of K, L, M, N, . . . with respect to one another?

15. Construct a scalene triangle ABC, and on the side of BC away from A, describe a triangle DBC, with DB= AB, and DC=AC. Join AD. What triangles in the figure are isosceles? What inference can you draw as to the angles BAC, BDC? Is any line in the figure bisected? What are the angles at the intersection of BC and AD? (Apply set-square.)