Note to the Teacher.—In all the development-lessons, the pupils are to be occupied with the diagrams, and not with the printed matter.
See Note A, Appendix.
Refer to Diagram 1, and show that
What are here drawn are intended to represent length only.
They have a little width, that they may be seen.
They are called lines.
A line is that which has length only.
Show that
Position is denoted by a point.
It occupies no space.
It has some size, that it may be seen.
The ends of a line are points.
A line may be regarded as a succession of points.
The intersection of two lines is a point.
A point is named by placing a letter near it.
Diagram 1.
A point may be represented by a dot. The point is in the center of the dot.
A point is that which denotes position only.
A line is named by naming the points at its ends.
Read all the lines in Diagram 1.
See Note B, Appendix.
Does the line m n change direction at the point 1?
At what other points does it change direction?
It is called a crooked line.
A crooked line is one that changes direction at some of its points.
The line o p changes direction at every point.
It is called a curved line.
A curved line is one that changes direction at every point.
Does the line i j change direction at any point?
It is called a straight line.
A straight line is one that does not change direction at any point.
The line q r winds about a line.
It is called a spiral line.
The line w x winds about a point.
It also is called a spiral line.
A spiral line is one that winds about a line or point.
The line 7 8[1] looks like waves.
1. To be read seven, eight, not seventy-eight.
It is called a wave line.
What kind of a line is a b?
Why? What is a straight line?
What kind of a line is 11 16?
Why? What is a crooked line?
What kind of a line is o p?
Why? What is a curved line?
What kind of a line is s t?
Why?
What kind of a line is 9 10?
Why? What is a spiral line?
What kind of a line is w x?
Why?
Read all the straight lines. (Diagram 2.)
Why is m n a straight line?
Define a straight line.
Read all the crooked lines.
Why is 7 8 a crooked line?
Define a crooked line.
Read all the curved lines.
Why is 5 6 a curved line?
What is a curved line?
Read all the wave lines.
Read all the spiral lines.
Why is 3 4 a spiral line?
Why is u v a spiral line?
What is a spiral line?
Diagram 2.
Diagram 3.
Let the pupils hold their books so that they will be straight up and down like the wall.
The straight line a b points to the center of the earth. (Diagram 3.)
It is called a vertical line.
Name all the vertical lines.
A vertical line is a straight line that points to the center of the earth.
The straight line o p points to the horizon.
It is called a horizontal line.
Read all the horizontal lines.
A horizontal line is a straight line that points to the horizon.
The line s t points neither to the center of the earth nor to the horizon.
It is called an oblique line.
Read all the oblique lines.
An oblique line is a straight line that points neither to the horizon nor to the center of the earth.
Note.—After going through with the lessons on angles, the pupils may be told that oblique lines are so called because they form oblique angles with the horizon.
Read all the vertical lines. (Diagram 4.)
Why is q r a vertical line?
What is a vertical line?
Read all the horizontal lines.
Why is 5 6 a horizontal line?
Define a horizontal line.
Read all the oblique lines.
Why is s t an oblique line.
What is an oblique line?
Note.—Lines that point in the same direction do not approach the same point.
Diagram 4.
Diagram 5.
Do the lines a b and c d (Diagram 5.) point in the same direction? (See note, page 15.)
Then they form an angle with each other.
What other line forms an angle with a b?
Which of the two lines c d, e f, has the greater difference of direction from the line a b?
Then which one forms the greater angle with a b?
What line forms a still greater angle with the line a b?
An angle is the difference of direction of two straight lines.
If the lines a b, e f, were made longer, would their direction be changed?
Then would there be any greater or less difference of direction?
Then would the angles formed by them be any greater or less?
Then does the size of an angle depend upon the length of the lines that form it?
If the lines a b, e f, were shortened, would the angle formed by them be any smaller?
If two lines form an angle with each other, and meet, the point of meeting is called the vertex.
What is the vertex of the angle formed by the lines k j, i j?—i j, i l?
An angle is named by three letters, that which denotes the vertex being in the middle. Thus, the angle formed by k j, i j, is read k j i, or i j k.
Read the four angles formed by the lines m n and o p.
The eight formed by r s, t u, and v w.
Read all the lines that form angles with the line a b. (Diagram 6.)
Which of them forms the greatest angle with it?
Diagram 6.
Which the least?
Of the two lines c d, g h, which forms the greater angle with e f?
Read all the angles whose vertices are at o on i j.
Which angle is the greater, l o m, or m o j?—i o k, or i o l?—l o j, or m o j?
Read all the angles formed by the lines v w and x y.
Read all the angles above the line n p.
Below the line n p. Above the line q r.
At the right of the line 5 u.
At the left. At the right of the line s t.
At the left of the line s t.
Which angle is the greater, n 1 3, or n 2 4?
If the lines x y and v w were lengthened or produced, would the angles v z x, y z w be any greater?
If they were shortened, would the angles be any less?
What is an angle?
Does the size of an angle depend upon the length of the lines which form it?
Diagram 7.
Are the angles a e c, c e b (Diagram 7.), on the same side of any line? What line?
By what other straight line are they both formed?
Then, because they are both on the same side of the same straight line a b, and are both formed by the second straight line c d, they are called “adjacent angles.”
The angles c e b, b e d are both on the same side of what straight line?
They are both formed by what second straight line?
Then what kind of angles are they?
Why are they called adjacent angles?
Read the adjacent angles below the line a b. Below the line c d.
How many pairs of adjacent angles can be formed by two straight lines?
Read all the adjacent angles formed by the lines l m and n p.
Are the angles a e c, b e d formed by the same straight lines?
Are they adjacent angles?
They are called “vertical angles.”
Vertical angles are angles formed by the same straight lines, but not adjacent to each other.
Read the other pair of vertical angles formed by the lines a b, c d.
Read all the vertical angles formed by the lines f g, i h. By l m, n p.
Why are the angles l o n, n o m adjacent angles?
Why are the angles l o n, p o m vertical angles?
Diagram 8.
Read the pairs of adjacent angles above the line a b. (Diagram 8.)
Why are they adjacent?
What are adjacent angles?
Read the adjacent angles below the line a b.
On the right of the line c d. On the left.
How many pairs of adjacent angles are formed by the intersection of two lines.
Read the pairs of adjacent angles formed by the lines f g and i h.
Read all the adjacent angles formed by the lines l m, n p.
Read all the pairs of vertical angles formed by the lines a b, c d.
Why are c e b and a e d called vertical angles?
What are vertical angles?
Read all the pairs of vertical angles formed by the lines h i, f g.
How many pairs of vertical angles are formed by the intersection of two lines?
Read all the pairs of vertical angles formed by the lines l m, n p.
What do we call the angles a o c, c o b? (Diagram 9.)
Are they equal to each other?
Then they are called right angles.
A right angle is one of two adjacent angles that are equal to each other.
Are the adjacent angles c o b, b o d equal to each other?
Then what are they called?
Read the right angles below the line a b. On the left of c d.
Read three right angles whose vertices are at p.
Diagram 9.
Is the angle m p q greater or less than the right angle m p r?
Then it is called an acute angle.
An acute angle is one which is less than a right angle.
Read four acute angles whose vertices are at p.
Acute means sharp.
Why is r p s an acute angle?
What is an acute angle?
Is the angle m p s greater or less than the right angle m p r?
Then it is called an obtuse angle.
An obtuse angle is one which is greater than a right angle.
What other obtuse angle has its vertex at p?
Obtuse means blunt.
Read three obtuse angles whose vertices are at x.
Acute and obtuse angles are also called oblique angles.
Read all the right angles formed by the lines a b and c d. (Diagram 10.)
Why are the adjacent angles c e b, b e d, right angles?
What is a right angle?
Read four right angles whose vertices are at n.
Which is the greater, the right angle p q r, or the right angle t s u?
Can one right angle be greater than another?
Read six acute angles whose vertices are at n.
Why is m n g an acute angle?
What is an acute angle?
Which is greater, the acute angle m n g, or the acute angle l n m?
May one acute angle be greater than another?
What three acute angles are equal to one right angle?
Diagram 10.
Which of the two acute angles v f w, y x z is the greater?
Read four obtuse angles whose vertices are at n.
Why is f n m an obtuse angle?
What is an obtuse angle?
What does obtuse mean? Acute?
By what other name are both called?
Which is greater, the large acute angle 1 4 2, or the small obtuse angle 1 4 3?
How much greater than the right angle is the obtuse angle f n l?
How much less than a right angle is f n i?
Diagram 11.
What kind of angles do the lines a b and c d make with each other? (Diagram 11.)
Then they are perpendicular to each other.
What line is perpendicular to x y?
Why is it perpendicular to it?
What line is perpendicular to z 1?
When is a line said to be perpendicular to another?
Can a line standing alone be properly called a perpendicular line?
What two lines are perpendicular to the lines r s?
Is the line g h perpendicular to the line i j? Why?
What other line is perpendicular to the line i j?
Read three lines that are perpendicular to the line a b.
Do the lines k l, m n, differ in direction? Then do they form any angle with each other?
They are said to be parallel to each other.
Read four other lines that are parallel with k l.
What line is parallel with 2 10?
Why?
Lines are parallel with each other when they do not differ in direction.
What kind of angles do the lines u t and 8 9 form with each other?
Then they are said to be oblique to each other.
Lines are oblique to each other when they form oblique angles.
See Note C, Appendix.
Diagram 12.
Read five lines that are perpendicular to the line a b. (Diagram 12.)
Five that are perpendicular to c d.
Two that are perpendicular to u v, and meet it. Three that do not meet it.
Why are o p and m n perpendicular to each other?
When are lines said to be perpendicular to each other?
Read four lines that are parallel with e f.
Why are the lines e f and g h said to be parallel to each other?
When are lines said to be parallel to each other?
Read four lines that are parallel to 5 6.
Four that are parallel to o p.
Is any line parallel to u v?
Can a single line be properly called perpendicular? Parallel?
If two lines are perpendicular to each other, what angle do they form?
If parallel, what angle? If oblique?
Diagram 13.
Is the angle a m n between the parallels, or outside of them? (Diagram 13.)
It is called an interior angle.
Read three other interior angles between the same parallels.
Why is b m n an interior angle?
An interior angle is one that lies between parallel lines.
Read the interior angles between the parallel lines g h and k l.
Why is o p l an interior angle?
What is an interior angle?
Is the angle a m e between the parallels, or outside of them?
It is called an exterior angle.
Read three other exterior angles formed by the lines a b, c d, and e f.
Why is the angle c n f an exterior angle?
An exterior angle is one that lies outside of the parallels.
Read all the interior angles formed by the lines a b, c d, and e f.
Why is m n d an interior angle?
What is an interior angle?
Read all the exterior angles formed by the same lines.
Why is d n f an exterior angle?
What is an exterior angle?
Read all interior angles formed by the lines g h, k l, and i j.
All the remaining interior angles in the diagram. All the exterior angles.
Diagram 14.
Are the angles e m b, b m n, on the same side of the intersecting line e f?
Are they adjacent?
Are e m b, m n d, on the same side of the intersecting line e f?
Are they adjacent?
Then they are called opposite angles.
Opposite angles lie on the same side of the intersecting line, but are not adjacent.
Are the angles e m b, f n d, on the same side of the intersecting line?
Are they adjacent?
Then are they opposite?
Are they interior or exterior angles?
Then they are “opposite exterior angles.”
Why are they exterior?
Why are they opposite?
Are the angles b m n, m n d, opposite angles?
Are they interior or exterior angles?
Then they are “opposite interior angles.”
Why are they opposite? Why interior?
Read the opposite exterior angles on the left of the line e f.
Read the opposite interior angles on the same side.
Are the opposite angles e m a, m n c, both exterior or interior?
Then they are opposite exterior and interior angles.
Read two pairs of opposite exterior and interior angles on the right of e f. On the left.
Do the angles b m n, m n c, lie on the same side of the intersecting line e f?
Are they adjacent to each other?
Are they vertical angles?
Then they are alternate angles.
Alternate angles lie on different sides of the intersecting line, and are neither adjacent nor vertical.
Are the alternate angles b m n, m n c, exterior or interior?
Then they are called “interior alternate angles.”
Read another pair of interior alternate angles between a b and c d.
Are the angles e m b, c n f, alternate angles? Why?
Are they exterior or interior?
Then what may they be called?
Read another pair of exterior alternate angles.
Why are e m a, d n f, alternate angles? Why exterior alternate?
Read the exterior opposite angles on the right of the line e f. (Diagram 14.)
On the left. On the right of r s. On the left.
Why are e m a, c n f, exterior angles?
Why are they opposite angles?
What are opposite angles?
Read the interior opposite angles on the right of the intersecting line e f.
On the left of it. On the right of r s. On the left.
Read the interior alternate angles formed by the lines a b, c d, and e f.
Which pair are acute angles?
Which pair are obtuse angles?
Why are b m n, m n c, interior angles? Why alternate? What are alternate angles?
Read the exterior alternate angles of the same lines.
Read the acute interior alternate angles of the parallels t u, v w. The obtuse.
The acute exterior alternate angles. Obtuse.
Read the pair of opposite exterior angles on the right of the line e f. On the left.
On the right of r s. On the left.
Diagram 15.
Read thirteen or more angles whose vertices are at c. (Diagram 15.)
Read four obtuse angles.
Read two right angles.
What three acute angles equal one right angle?
Which is greater, the right angle 4, or the right angle 5?
The obtuse angle 6, or the acute angle 7?
Read twelve pairs of adjacent angles formed by the lines w x, &c.
Read six pairs of vertical angles formed by the same lines.
Read all the interior angles formed by the lines i j, k l, and m n.
Read all the exterior angles formed by the same lines.
Two pairs of opposite exterior angles.
Two pairs of opposite interior angles.
Four pairs of opposite exterior and interior angles.
Two pairs of alternate interior angles.
Two pairs of alternate exterior angles.
Why are the angles i o m, m o j, called adjacent?
What are adjacent angles?
What kind of an angle is i o m? Why?
What is an acute angle?
What kind of an angle is m o j? Why?
What is an obtuse angle?
Why are a c f, f c b, right angles?
What is a right angle?
Why are m o i, j o p, vertical angles?
What are vertical angles?
Why is m o i an exterior angle?
What is an exterior angle?
Why is j o p an interior angle?
What is an interior angle?
Why are m o i, o p k, opposite angles?
What are opposite angles?
Why are j o p, o p k, alternate angles?
What are alternate angles?