Of Angles -Right, Acute, and Obtuse.

the vertex in the middle. Thus, we say the angle CAB. We may, however, say simply, the angle A.

2. One line is perpendicular to another, when it inclines no more to the one side than to the other. The angles on each side are then equal

A

B

to each other. Thus, if the line DB is perpendicular to AC, the angle DBA is equal to DBC.

3. When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC are right an

D

A

B

gles. Hence, all right angles are equal to each other.

4. An acute angle is less than a right angle. Thus, DBC is an acute angle.

5. An obtuse angle is greater than a right angle. Thus, EBC is an obtuse angle.

D

B

E

B

QUEST.-2. When is one line perpendicular to another? Are the angles on each side then equal? 3. When two lines are perpendicular to each other, what are the angles on each side called? Are all right angles equal to each other? Make two right angles. Point out the equal angles. 4. What is an acute angle? Make one. 5. What is an obtuse angle? Make one.

Measure.-Of Angles.

6. The circumference of a circle is a curve line, all the points of which are equally distant from a certain point within, called the centre. Thus, if all the points of the curve AEB are equally distant from the centre C, this

curve will be the circumference of a circle.

7. The circumference of a circle is used for the measurement of angles. For this purpose it is divided into 360 equal parts, called degrees, each degree is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds, are marked thus, °, ', "; and 9° 18' 10", are read, 9 degrees, 18 minutes, and 10 seconds.

8. Suppose the circumference of a circle to be divided into 360 equal parts, beginning at the point B. If, through the point of division marked 40, we draw CE, then, the angle ECB will be equal to 40 degrees. If

180

90 80

F

40

E

B

QUEST.-6. What is the circumference of a circle? Make one, and point out the centre. 7. For what is the circumference of a circle used? Into how many parts is it supposed to be divided? What is each part called? Into how many parts is each degree divided? Each minute into how many parts? Show how the degrees, minutes, and seconds are marked. 8. Explain how an angle is measured on the circumference of a circle. Draw a line which shall make with another line an angle of 30.- Then a line which shall make an angle of 60°.

Of Angles.

we draw CF through the point of division marked 80, it will make with CB an angle equal to 80 degrees.

9. If two lines AB, DE, are perpendicular to each other, the four angles BCD, DCA, ACE, and ECB will be equal. These two lines will divide the circumference of the circle into the four equal parts BD, DA,

D

B

AE, and EB, and each part will measure one of the right angles. But one quarter of 360 degrees, is 90 degrees. Hence, one right angle contains 90 degrees, two right angles 180 degrees, three right angles 270 degrees, and four right angles 360 degrees.

10. One quarter of the circumference is called a quadrant, and contains 90 degrees. One half of the circumference is called a semi-circumference,

and contains 180 degrees. Thus, AC is a quadrant, and ACB is a semi-circumference.

QUEST.-9. If two lines are perpendicular to each other, how many right angles will be formed? Into how many equal parts will these lines divide the circumference? How many degrees does one right angle contain? How many degrees in two right angles? In three right angles? In four right angles? 10. What is one quarter of the circumference called? How many degrees does it contain? What is half the circumference called? How many degrees does it contain?.

Of Angles.

11. If a straight line EB meets another straight line AC, the sum of the angles ABE and EBC, will be

E

equal to two right angles, since these A

two angles are measured by half the circumference.

And if there be several angles CBF, FBE, EBD, DBA, formed on the

F

same side of a line, their sum for a c

E

D

like reason, will be equal to two right angles.

12. The sum of all the angles ACB, BCD, DCA, which can be formed about any point as C, is equal to four right angles, or 360 degrees, since they are measured by the entire circumference.

13. If two lines intersect each other, the opposite angles A and A are called vertical angles. These angles are equal to each other, and so also, are the opposite angles B and B.

B

B

QUEST.-11. If one straight line meets another, what is the sum of the two angles on the same side equal to ? What is the sum of several angles formed on the same side of a straight line equal to ? 12. What is the sum of all the angles which can be formed about the same point equal to ? 13. If two straight lines intersect each other, what are the opposite angles called? Are these angles equal or unequal?

Of Lines.

SECTION III.

OF PARALLEL, OBLIQUE, AND PERPENDICULAR LINES.

1. Two straight lines are said to be parallel when they are at the same distance from each other at

every point. Parallel lines will never meet each other.

2. Two curves are said to be par

allel or concentric, when they are at the same distance from each other. Parallel curves will not meet each other.

3. Oblique lines are those which approach each other, and meet if sufficiently prolonged.

4. Lines which are parallel to the horizon, or to the water level, are called horizontal lines. Thus, the eaves of a house are horizontal.

QUEST.—1. When are two straight lines said to be parallel? Will parallel lines meet each other? 2. When are two curves said to be parallel? Will parallel curves meet each other? 3. What are oblique lines? Make two oblique lines. 4. What are horizontal lines? Are the eaves of a house horizontal ?