PART FIRST.
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 NoteA, Appendix.
Refer toDiagram1, and show that
What are here drawn are intended to representlengthonly.
They have a little width, that they may be seen.
They are calledlines.
A line is that which has length only.
Show that
Position is denoted by a point.
It occupies no space.
It hassomesize, 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.
Diagram 1.
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 Diagram1.
See NoteB, Appendix.
Does the linemnchange 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 atsomeof its points.
CURVED LINES.
The lineo pchanges direction at every point.
It is called a curved line.
A curved line is one that changes direction ateverypoint.
Does the linei jchange direction at any point?
It is called a straight line.
A straight line is one that doesnotchange direction at any point.
The lineq rwinds about a line.
It is called aspiral line.
The linew xwinds 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.
1.To be read seven, eight, not seventy-eight.
It is called a wave line.
What kind of a line isa 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 iso p?
Why? What is a curved line?
What kind of a line iss t?
Why?
What kind of a line is 9 10?
Why? What is a spiral line?
What kind of a line isw x?
Why?
Read all the straight lines. (Diagram2.)
Why ism na 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 isu va spiral line?
What is a spiral line?
Diagram 2.
Diagram 2.
Diagram 2.
Diagram 3.
Diagram 3.
Diagram 3.
Let the pupils hold their books so that they will be straight up and down like the wall.
The straight linea bpoints to the center of the earth. (Diagram3.)
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 lineo ppoints 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 lines tpoints 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. (Diagram4.)
Why isq ra 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 iss tan oblique line.
What is an oblique line?
Note.—Lines that point in the same direction do not approach the same point.
Diagram 4.
Diagram 4.
Diagram 4.
Diagram 5.
Diagram 5.
Diagram 5.
Do the linesa bandc d(Diagram5.) point in the same direction? (See note, page15.)
Then they form ananglewith each other.
What other line forms an angle witha b?
Which of the two linesc d,e f, has the greater difference of direction from the linea b?
Then which one forms the greater angle witha b?
What line forms a still greater angle with the linea b?
An angle is the difference of direction of two straight lines.
If the linesa 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 thesizeof an angle depend upon the length of the lines that form it?
If the linesa 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 linesk 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 byk j,i j, is readk j i, ori j k.
Read the four angles formed by the linesm nando p.
The eight formed byr s,t u, andv w.
Read all the lines that form angles with the linea b. (Diagram6.)
Which of them forms the greatest angle with it?
Diagram 6.
Diagram 6.
Diagram 6.
Which the least?
Of the two linesc d,g h, which forms the greater angle withe f?
Read all the angles whose vertices are atooni j.
Which angle is the greater,l o m, orm o j?—i o k, ori o l?—l o j, orm o j?
Read all the angles formed by the linesv wandx y.
Read all the angles above the linen p.
Below the linen p. Above the lineq r.
At the right of the line 5u.
At the left. At the right of the lines t.
At the left of the lines t.
Which angle is the greater,n1 3, orn2 4?
If the linesx yandv wwere lengthened or produced, would the anglesv z x,y z wbe 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.
Diagram 7.
Diagram 7.
Are the anglesa e c,c e b(Diagram7.), 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 linea b, and are both formed by the second straight linec d, they are called “adjacent angles.”
The anglesc e b,b e dare 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 linea b. Below the linec d.
How many pairs of adjacent angles can be formed by two straight lines?
Read all the adjacent angles formed by the linesl mandn p.
Are the anglesa e c,b e dformed 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 linesa b,c d.
Read all the vertical angles formed by the linesf g,i h. Byl m,n p.
Why are the anglesl o n,n o madjacent angles?
Why are the anglesl o n,p o mvertical angles?
Diagram 8.
Diagram 8.
Diagram 8.
Read the pairs of adjacent angles above the linea b. (Diagram8.)
Why are they adjacent?
What are adjacent angles?
Read the adjacent angles below the linea b.
On the right of the linec 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 linesf gandi h.
Read all the adjacent angles formed by the linesl m,n p.
Read all the pairs of vertical angles formed by the linesa b,c d.
Why arec e banda e dcalled vertical angles?
What are vertical angles?
Read all the pairs of vertical angles formed by the linesh 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 linesl m,n p.
What do we call the anglesa o c,c o b? (Diagram9.)
Are they equal to each other?
Then they are calledright angles.
A right angle is one of two adjacent angles that are equal to each other.
Are the adjacent anglesc o b,b o dequal to each other?
Then what are they called?
Read the right angles below the linea b. On the left ofc d.
Read three right angles whose vertices are atp.
Diagram 9.
Diagram 9.
Diagram 9.
Is the anglem p qgreater or less than the right anglem p r?
Then it is called anacute angle.
An acute angle is one which is less than a right angle.
Read four acute angles whose vertices are atp.
Acute means sharp.
Why isr p san acute angle?
What is an acute angle?
Is the anglem p sgreater or less than the right anglem p r?
Then it is called anobtuse angle.
An obtuse angle is one which is greater than a right angle.
What other obtuse angle has its vertex atp?
Obtuse means blunt.
Read three obtuse angles whose vertices are atx.
Acute and obtuse angles are also called oblique angles.
Read all the right angles formed by the linesa bandc d. (Diagram10.)
Why are the adjacent anglesc e b,b e d, right angles?
What is a right angle?
Read four right angles whose vertices are atn.
Which is the greater, the right anglep q r, or the right anglet s u?
Can one right angle be greater than another?
Read six acute angles whose vertices are atn.
Why ism n gan acute angle?
What is an acute angle?
Which is greater, the acute anglem n g, or the acute anglel n m?
May one acute angle be greater than another?
What three acute angles are equal to one right angle?
Diagram 10.
Diagram 10.
Diagram 10.
Which of the two acute anglesv f w,y x zis the greater?
Read four obtuse angles whose vertices are atn.
Why isf n man 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 anglef n l?
How much less than a right angle isf n i?
Diagram 11.
Diagram 11.
Diagram 11.
What kind of angles do the linesa bandc dmake with each other? (Diagram11.)
Then they are perpendicular to each other.
What line is perpendicular tox y?
Why is it perpendicular to it?
What line is perpendicular toz1?
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 linesr s?
Is the lineg hperpendicular to the linei j? Why?
What other line is perpendicular to the linei j?
Read three lines that are perpendicular to the linea b.
Do the linesk l,m n, differ in direction? Then do they form any angle with each other?
They are said to beparallelto each other.
Read four other lines that are parallel withk 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 linesu tand 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 NoteC, Appendix.
Diagram 12.
Diagram 12.
Diagram 12.
Read five lines that are perpendicular to the linea b. (Diagram12.)
Five that are perpendicular toc d.
Two that are perpendicular tou v, and meet it. Three that do not meet it.
Why areo pandm nperpendicular to each other?
When are lines said to be perpendicular to each other?
Read four lines that are parallel withe f.
Why are the linese fandg hsaid 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 too p.
Is any line parallel tou 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.
Diagram 13.
Diagram 13.
Is the anglea m nbetween the parallels, or outside of them? (Diagram13.)
It is called aninterior angle.
Read three other interior angles between the same parallels.
Why isb m nan interior angle?
An interior angle is one that lies between parallel lines.
Read the interior angles between the parallel linesg handk l.
Why iso p lan interior angle?
What is an interior angle?
Is the anglea m ebetween the parallels, or outside of them?
It is called anexterior angle.
Read three other exterior angles formed by the linesa b,c d, ande f.
Why is the anglec n fan exterior angle?
An exterior angle is one that lies outside of the parallels.
Read all the interior angles formed by the linesa b,c d, ande f.
Why ism n dan interior angle?
What is an interior angle?
Read all the exterior angles formed by the same lines.
Why isd n fan exterior angle?
What is an exterior angle?
Read all interior angles formed by the linesg h,k l, andi j.
All the remaining interior angles in the diagram. All the exterior angles.
Diagram 14.
Diagram 14.
Diagram 14.
Are the anglese m b,b m n, on the same side of the intersecting linee f?
Are they adjacent?
Aree m b,m n d, on the same side of the intersecting linee 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 anglese 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 anglesb 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 linee f.
Read the opposite interior angles on the same side.
Are the opposite anglese m a,m n c, both exterior or interior?
Then they areopposite exterior and interior angles.
Read two pairs of opposite exterior and interior angles on the right ofe f. On the left.
Do the anglesb m n,m n c, lie on the same side of the intersecting linee 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 anglesb m n,m n c, exterior or interior?
Then they are called “interior alternate angles.”
Read another pair of interior alternate angles betweena bandc d.
Are the anglese 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 aree m a,d n f, alternate angles? Why exterior alternate?
Read the exterior opposite angles on the right of the linee f. (Diagram14.)
On the left. On the right ofr s. On the left.
Why aree 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 linee f.
On the left of it. On the right ofr s. On the left.
Read the interior alternate angles formed by the linesa b,c d, ande f.
Which pair are acute angles?
Which pair are obtuse angles?
Why areb 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 parallelst u,v w. The obtuse.
The acute exterior alternate angles. Obtuse.
Read the pair of opposite exterior angles on the right of the linee f. On the left.
On the right ofr s. On the left.
Diagram 15.
Diagram 15.
Diagram 15.
Read thirteen or more angles whose vertices are atc. (Diagram15.)
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 linesw x, &c.
Read six pairs of vertical angles formed by the same lines.
Read all the interior angles formed by the linesi j,k l, andm 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 anglesi o m,m o j, called adjacent?
What are adjacent angles?
What kind of an angle isi o m? Why?
What is an acute angle?
What kind of an angle ism o j? Why?
What is an obtuse angle?
Why area c f,f c b, right angles?
What is a right angle?
Why arem o i,j o p, vertical angles?
What are vertical angles?
Why ism o ian exterior angle?
What is an exterior angle?
Why isj o pan interior angle?
What is an interior angle?
Why arem o i,o p k, opposite angles?
What are opposite angles?
Why arej o p,o p k, alternate angles?
What are alternate angles?