116. How many angles are formed on each face of a cube by its boundary lines?

117. Find the number of degrees in the sum of all the angles formed on the faces of a cube by their boundary lines.

118. How many square inches are there in the surfaces of a box whose height is 4 inches, its width twice its height, and its length 3 times its width?

119. How many square inches are there in the surfaces. of a brick which is 9 inches long, as wide as long, and as thick as wide?

CHAPTER VI.

TRIANGLES AND LINES.

1. Draw a line 5 inches long. With one end of the line as a center and a radius of 5 inches, describe an arc. With the other end

as a center and the same radius, describe an

arc intersecting the first. Join the point of

intersection with the ends of the line. What kind of a plane figure is thus formed?

NOTE. A triangle whose sides are all equal is called an Equilateral Triangle.

2. How many centimeters are there in the perimeter of an equilateral triangle, one side of which is 8 millimeters?

3. Find the length in decimeters of one side of an equilateral triangle whose perimeter is 48 centimeters.

4. Find the difference between the length of the perimeter of an equilateral triangle each of whose sides is 7 inches, and that of an equilateral rectangle each of whose sides is 7 inches.

QUERY. What does "equilateral" mean?

5. The perimeter of a triangle whose sides are each 6 centimeters is what fractional part of the perimeter of a hexagon each of whose sides is 6 centimeters?

6. Draw two equal lines making any angle, and join their extremities. What plane figure is formed?

NOTE. A triangle which has two of its sides equal is called an Isosceles Triangle. The unequal side is called the Base.

7. Find the base of an isosceles triangle whose perimeter is 90 millimeters and each of whose equal sides is 35 millimeters.

8. Find each of the equal sides of an isosceles triangle whose perimeter is 40 inches and base 10 inches.

9. Each of the equal sides of an isosceles triangle is double the base, and the perimeter is 45 centimeters. Find each side.

10. The base of an isosceles triangle is 5 inches longer than each of its equal sides, and the perimeter is 35 inches. Find each side.

QUERY. - What shall x equal?

11. The sum of the equal sides of an isosceles triangle is 4 times the base, and the perimeter is 15 inches. Find each side.

12. Cut out an isosceles triangle, fold it so that the equal sides shall coincide, and show the truth of the following theorem:

PRINCIPLE 17. In an isosceles triangle the angles opposite the equal sides are equal.

13. In the isosceles triangle ABC the angle BAC is 70°. How many degrees are there in each exterior angle formed by prolonging the side AC? Quote principle.

B

Д

NOTE.

An angle formed by prolonging one side of a polygon is called an Exterior Angle.

14. The exterior angle DBC, formed by prolonging one leg of the isosceles triangle ABC, is 115°. Find each of the base angles and the exterior angle BCE.

D

B

A

15. The exterior angle formed by prolonging the base of an isosceles triangle contains 3 times as many degrees as the interior base angle. How many degrees are there in the sum of the base angles? How many in their difference?

16. The exterior angle formed by prolonging one of the legs of an isosceles triangle is 26° more than a base angle. How many degrees are there in each base angle?

17. Draw and cut out an equilateral triangle. Fold it so that two of its equal sides shall coincide. Are the angles opposite them equal? Smooth it out and fold it so that another pair of equal sides shall coincide. Are the angles all equal?

NOTE.

A triangle whose angles are all equal is Equiangular. PRINCIPLE 18. An equilateral triangle is equiangular.

18. In the equilateral triangle ABC, of which BC is the base, which is the greater, the vertical angle or one of the base angles?

NOTE.

Any side of a triangle on which it may be supposed to stand may be called the Base, and the angle opposite the base is called the Vertical Angle.

19. If AB is the base of the triangle ABC, what is the vertical angle? If Bis the vertical angle, what is the base?

B

20. Draw three equilateral triangles of the same dimensions and place them so that they have the vertex of an angle of each at a common point, as at

C in the figure. It will be seen that a straight line is formed by the bases of the outer triangles.

The straight line ACE is what fractional part of the broken line ABDE?

B

E

21. How many degrees are there in the sum of the angles at C? Quote geometric principles.

22. How many degrees are there in each of the angles at C? Give reasons.

23. How many degrees are there in angle BAC? In angle ABC? How many are there in each angle of each equilateral triangle ? How many in the sum of the angles of either one of the equilateral triangles?

24. How many degrees are there in

the angle x, formed by prolonging a side of an equilateral triangle?

25. Draw an equilateral triangle and prolong its sides so as to form an exterior angle at each vertex. Cut out the exterior angles and place them around a common point. Will they form a continuous surface around the point? How many degrees are there in their sum?

26. Draw a regular polygon of three sides and name it.

NOTE. When a polygon is equilateral and equiangular it is called a Regular Polygon.

27. Draw a regular polygon of four sides and write its name upon it. How many right angles has it? How many degrees are there in all its angles?