PART SECOND.AXIOMS AND THEOREMS.

The triangle A is equal to the triangle C.

The triangle B is also equal to the triangle C.

What do you think of the two triangles A and B? Why?

If two things are separately equal to the same thing, they are equal to each other.

The square A is equal to the square B.

To the rectangle C add the square A, and we have an L pointing in what direction?

To the same rectangle C add the square B, and we have an L pointing in what direction?

Which is larger, the L pointing to the left, or that pointing to the right?

To what same thing did you add two equals?

What two equals did you add to it?

What was the first sum?

The second?

What do you think of the two sums?

If equals be added to the same thing, the sums will be equal.

The square A is equal to the square B.

From the inverted T take away the square A, and we have an L pointing in what direction?

From the same Fig. T take away the square B, and we have an L pointing in what direction?

Which is larger, the L pointing to the right, or that pointing to the left?

What two equal things did we take away from the same thing?

From what same thing did we take them away?

What did we find true of the two remainders?

If equals be taken from the same thing, the remainders will be equal.

Axiom 4.

The rectangle 1 2 is equal to the rectangle 1 3.

From the rectangle 1 2 take away the square A, and what rectangle remains?

From the rectangle 1 3 take away the same square A, and what rectangle remains?

Which is greater, the rectangle B, or the rectangle C?

What same thing did we take away from equals?

From what did we first take it?

What remained?

From what did we next take it?

What remained?

What did we find true of the two remainders?

If the same thing be taken from equals, the remainders will be equal.

If equals be added to equals, the sums will be equal.

If equals be subtracted from equals, the remainders will be equal.

If the halves of two things are equal, the wholes will be equal.

Axiom 8.

Every Whole is equal to the sum of all its parts.

From one point to another only one straight line can be drawn.

A straight line is the shortest distance between two points.

If two things coincide throughout their whole extent, they are equal.

Diagram 29.

Do the angles Blue, Red, take up all the space on the linea b?

Do the angles Blue, Yellow, Red, take up all the space on the line?

Do the angles Blue, Yellow, Green, Red, take up all the space on the line?

Is there room between any two of the angles to put in another angle?

Then are not the angles Blue, Yellow, Green, Red, equal to all the space on the linea b?

Note.—The wordspace, as here used, meansangular space; and it is indispensable that the teacher impress this fact upon the learner.

By means of former lessons, the pupil has learned positively, that an angle is the difference between the directions of two lines; and, impliedly, that the included space has nothing to do with the size of the angle. There cannot, therefore, be much danger that the pupil will imbibe any erroneous notion from this style of expression, which is very much more simple than to say that the difference of direction of two given lines is equal to the difference of direction of two other given lines, which style will be used somewhat later in these lessons.

Diagram 30.

Are the adjacent angles Green, Red, equal to all the angular space on the linea b?

Place a paper square corner or right angle on the linea bat theleftofc dwith its vertex atc.

It will cover all the angle Green and part of the angle Red up to the linec d.

Now place another square corner on the linea bto therightof the linec d, and with its vertex at the pointc.

It will cover the remaining part of the angle Red, and two edges of the square corners will meet along the linec d.

Are the two right angles equal to all the angular space on the linea b?

Then if the two adjacent angles Green, Red, are equal to all the angular space on the linea b, and the two right angles are also equal to thesame space, what do you infer concerning theadjacent anglesand thetwo right angles?

What axiom do you apply when you say that theadjacentangles are equal to thetwo right angles?

To whatsame thingdid you find two things separately equal?

What did you first see equal to it?

What did you next see equal to it?

Then what did youfindtrue?

If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?

Then,—

Any two adjacent angles are equal to two right angles.

If we draw the straight linec dwhere the edges of the square corners come together, what kind of angles willa c d,d c b, be?

See now if you can understand the following demonstration:—

We wish to prove that

Any two adjacent angles are equal to two right angles.

Let the two straight linesa b,m n, intersect each other in the pointc. (Diagram30.)

Then will any two adjacent angles, as Green, Red, be equal to two right angles?

For, from the pointc, draw the straight linec dso as to make the anglesa c d,d c b, right angles.

The adjacent angles Green, Red, are equal to all the angular space on the linea b.

The right anglesa c d,d c b, are also equal to all the angular space on the linea b.

Therefore the adjacent angles Green, Red, are equal to two right angles.

To what same thing did you find two things equal?

What did you first see equal to it?

What did you next see equal to it?

Then what new thing did you find true?

What axiom did you make use of?

Diagram 31.

By means of Fig. A,—

1. Prove that the adjacent angles Green, Red, are equal to two right angles.

2. Prove that the adjacent angles Blue, Yellow, are equal to two right angles.

By means of Fig. B,—

3. Prove that the adjacent angles Green, Red, are equal to two right angles.

4. Prove that the adjacent angles Yellow, Blue, are equal to two right angles.

By means of Fig. C,—

5. Prove that the adjacent angles Red, Blue, are equal to two right angles.

6. Prove that the adjacent angles Green, Yellow, are equal to two right angles.

7. Give the preceding demonstrations again, but name the angles by their letters instead of by their colors.

Diagram 32.

TEST LESSON.

By means of Fig. A prove,—

1. That the adjacent anglesa c m,m c b, are equal to two right angles.

2. That the adjacent anglesa c n,n c b, are equal to two right angles.

By means of Fig. B prove,—

3. That the adjacent anglesa c n,n c b, are equal to two right angles.

4. That the adjacent anglesa c m,m c b, are equal to two right angles.

By means of Fig. C prove,—

5. That the adjacent anglesa c m,m c b, are equal to two right angles.

6. That the adjacent anglesa c n,n c b, are equal to two right angles.

By means of Fig. D prove,—

7. That the adjacent anglesa c n,n c b, are equal to two right angles.

8. That the adjacent anglesb c m,m c a, are equal to two right angles.

Diagram 33.

What kind of angles are P and S?

How do the adjacent angles Yellow, Blue, compare with the right angles P, S?

How do the adjacent angles Blue, Red, compare with the two right angles?

Then if the adjacent angles Yellow, Blue, are equal to two right angles, and the adjacent angles Blue, Red, are also equal to two right angles, what do you think of the two pairs of adjacent angles, Yellow, Blue, and Blue, Red?

If, from the adjacent angles Yellow, Blue, we take away the angle Blue, what remains?

If, from the adjacent angles Blue, Red, we take away the same angle Blue, what remains?

Then, since the same angle Blue has been taken from equal pairs of adjacent angles, what do you think of the two remainders, Yellow, Red?

Suppose the linesa bandm nwere so drawn that the angles Yellow, Red, were larger or smaller, would they still be equal to each other?

Then,—

All vertical angles are equal to each other.

Diagram 34.

We wish to prove that

All vertical angles are equal to each other.

Let the straight linesa b,m n, intersect each other at the pointc, then will any two vertical angles, as Yellow, Red, be equal to each other.

For the adjacent angles Yellow, Blue, are equal to two right angles.[3]

3.When this comparison is made, let the pupil look at the right angles P and S.

The adjacent angles Blue, Red, are also equal to two right angles.

Therefore the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red.

If, from the adjacent angles Yellow, Blue, we take away the angle Blue, we shall have left the angle Yellow.

If, from the adjacent angles Blue, Red, we take away the same angle Blue, we shall have left the angle Red.

Therefore the vertical angles Yellow, Red, are equal to each other.

When you say that the adjacent angles Yellow, Blue, are equal to two right angles, do you know it because youseeit, or because you haveprovedit?

How do you know that the adjacent angles Blue, Red, are equal to two right angles?

When you say the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red, what axiom do you use?

What same thing do you take away from equals?

From what equals do you take it away?

When you take the angle Blue from the adjacent angles Yellow, Blue, what is the remainder?

When you take the same angle Blue from the adjacent angles Blue, Red, what is the remainder?

What do you find true of the two remainders?

What axiom do you use?

Diagram 35.

The adjacent angles Yellow, Green, are equal to what?

The adjacent angles Green, Red, are equal to what?

Then what do you know of the two pairs of adjacent angles Yellow, Green, and Green, Red?

From the adjacent angles Yellow, Green, take away the angle Green. What remains?

From the adjacent angles Green, Red, take the same angle Green. What remains?

What do you know of the two remainders?

Why?

What axiom do you use?

In the last lesson, when you proved the vertical angles Yellow, Red, equal to each other, you made use of the angle Blue; now prove the same two angles equal by means of the angle Green.

The adjacent angles Blue, Red, are equal to what?

The adjacent angles Red, Green, are equal to what?

Then what do you know of the two pairs of adjacent angles Blue, Red, and Red, Green?

From the adjacent angles Blue, Red, take away the angle Red. What remains?

From the adjacent angles Red, Green, take away the same angle Red. What remains?

Then what do you know of the two remainders, Blue, Green?

Now apply the preceding demonstration to the vertical angles Blue, Green.

Prove the vertical angles Blue, Green, equal to each other by means of the angle Yellow.

Diagram 36.

By means of Fig. A,—

1. Prove that the vertical angles Yellow, Red, are equal to each other, using the angle Green.

2. Prove the same thing, using the angle Blue.

3. Prove that the vertical angles Blue, Green, are equal to each other, using the angle Yellow.

4. Prove the same thing, using the angle Red.

By means of Fig. B,—

5. Prove the vertical angles Yellow, Red, equal to each other, using the angle Green.

6. Prove the same thing, using the angle Blue.

7. Prove the vertical angles Green, Blue, equal by means of the angle Red.

8. Prove the same thing by means of the angle Yellow.

Go through the preceding eight demonstrations again, calling the angles by their letters instead of by their colors.

By means of Fig. C, prove that

9.a c nequalsm c b, by means ofa c m.

10.a c nequalsm c b, by means ofb c n.

11.a c mequalsn c b, by means ofa c n.

12.a c mequalsn c b, by means ofm c b.

By means of Fig. D, prove that

13.m c aequalsb c n, by means ofa c n.

14.m c aequalsb c n, by means ofm c b.

15.m c bequalsa c n, by means ofm c a.

16.m c bequalsa c n, by means ofb c n.

Diagram 37.

In the above diagram, the linesa b,c d, are parallel, and are intersected by the linee fat the pointsmandn.

The angle Red measures the difference of direction between the linem band what other line?

The angle Yellow measures the difference of direction between the linen dand what other line?

Then, as the linesm bandn dare parallel, must there not be the same difference of direction between them and the linee f?

Then can there be any difference between the angles which measure those equal directions?

Then what do you think of the opposite exterior and interior angles Red, Yellow?

DEMONSTRATION.

We wish to prove that

Opposite exterior and interior angles are equal to each other.

Let the straight linee fintersect the two parallel straight linesa b,c d, at the pointsmandn.

Then will any two opposite exterior and interior angles, as Red, Yellow, be equal to each other.

For the angle Red measures the difference of direction of the linesm bande f.

And the angle Yellow measures the difference of direction of the linesn dande f.

But because the linesm b,n d, are parallel, these differences are equal.

Therefore the angles which measure them are equal; that is,

The opposite exterior and interior angles Red, Yellow, are equal to each other.

Diagram 38.

By means of Fig. A,—

1. Prove that the opposite exterior and interior angles Green, Blue, are equal to each other.

2. Prove that the opposite exterior and interior angles Red, Yellow, are equal to each other.

3. Prove the opposite exterior and interior anglesc n e,a m n, equal.

4. Prove the opposite exterior and interior anglese n d,n m b, equal.

By means of Fig. B,—

5. Prove the opposite exterior and interior anglese m a,m n d, equal.

6. Prove the opposite exterior and interior anglesa m n,d n f, equal.

7. Prove the opposite exterior and interior anglese m b,m n c, equal.

8. Prove the opposite exterior and interior anglesb m n,c n f, equal.

Diagram 39.

What do you know of the opposite exterior and interior angles Red, Yellow?

What do you know of the vertical angles Red, Green?

Then if the interior alternate angles Green, Yellow, are separately equal to the angle Red, what new fact do you know?

What axiom do you employ?

To what same thing did you find two things equal?

What two things did you find equal to it?

DEMONSTRATION.

We wish to prove that

Any two interior alternate angles are equal to each other.

Let the straight linee fintersect the two parallel straight linesa b,c d, in the pointsmandn.

Then will any two interior alternate angles, as Green, Yellow, be equal to each other.

For the opposite exterior and interior angles Red, Yellow, are equal.

The vertical angles Red, Green, are also equal.

Then because the interior alternate angles Green, Yellow, are separately equal to the angle Red, they are equal to each other.

Diagram 40.

What do you know of the vertical angles Green, Red, in Fig. A?

What do you know of the opposite exterior and interior angles Red, Yellow?

Then if the interior alternate angles Green, Yellow,are separately equal to the angle Red, what do you infer?

By means of Fig. A,—

1. Prove that the interior alternate angles Green, Yellow, are equal, using the angle Red.

2. Prove the same angles equal, using the angle Blue.

3. Go through the same demonstrations again, calling the angles by their letters instead of by their colors.

By means of Fig. B,—

4. Prove the interior alternate angles Red, Blue, equal, using the angle Yellow.

5. Prove the same angles equal, using the angle Green.

6. Go through the same two demonstrations again, naming the angles by their letters instead of by their colors.

By means of Fig. C,—

7. Prove the interior alternate anglesc n m,n m b, equal, using the anglef n d.

8. Prove the same, using the anglea m e.

9. Prove the interior alternate anglesa m n,m n d, equal, using the anglee m b.

10. Prove the same, using the anglec n f.

Diagram 41.

What do you know of the opposite exterior and interior angles Red, Yellow?

What do you know of the vertical angles Yellow, Green?

Then if the exterior alternate angles Red, Green, are separately equal to the angle Yellow, what new thing do you know to be true?

What axiom do you employ?

To what same thing did you know two things to be equal?

What two things did you know to be equal to it?

Then what new thing did youfindto be true?

DEMONSTRATION.

We wish to prove that

Any two exterior alternate angles are equal to each other.

Let the straight linee fintersect the two parallel straight linesa b,c d, at the pointsmandn.

Then will any two exterior alternate angles, as Red, Green, be equal.

For the opposite exterior and interior angles Red, Yellow, are equal to each other.

And the vertical angles Yellow, Green, are also equal to each other.

Then because the exterior alternate angles Red, Green, are separately equal to the angle Yellow, they are equal to each other.

Diagram 42.

What do you know of the opposite exterior and interior angles Yellow, Red?

What do you know of the vertical angles Red, Blue?

Then if the exterior alternate angles Yellow, Blue, are separately equal to the angle Red, what do you know of them?

By means of Fig. A,—

1. Prove that the exterior alternate angles Yellow, Blue, are equal, using the angle Red.

2. Prove the same thing, using the angle Green.

3. Go through the same demonstrations, calling the angles by their letters.

4. Prove the exterior alternate anglese m b,c n f, equal, using the anglea m n.

5. Prove the same, using the anglem n d.

By means of Fig. B,—

6. Prove that the exterior alternate anglesc m e,f n b, are equal, using the anglen m d.

7. Prove the same, using the anglea n m.

8. Prove the exterior alternate anglese m d,a n f, equal, using the anglec m n.

9. Prove the same, using the anglem n b.

Diagram 43.

What do you know of the interior alternate angles Yellow, Red?

If to the angle Green you add the angle Yellow, what is the sum?

If to the same angle Green you add the equal angle Red, what is the sum?

Then, having added equals to the same thing, what do you think of the two sums,—the adjacent angles Green, Yellow, and the interior opposite angles Green, Red?

What do you know of the adjacent angles Green, Yellow, and the right angles P, S?

Then if the interior opposite angles Green, Red, and the two right angles P, S, are separately equal to the adjacent angles Green, Yellow, what new thing do you know?

We wish to prove that

Any two interior opposite angles are equal to two right angles.

Let the straight linee fintersect the two parallel straight linesa b,c d, in the pointsmandn.

Then will any two interior opposite angles be equal to two right angles.

For the interior alternate angles Yellow, Red, are equal.

If to the angle Green we add the angle Yellow, we shall have the adjacent angles Green, Yellow.

If to the same angle Green we add the equal angle Red, we shall have the interior opposite angles Green, Red.

Then the adjacent angles Green, Yellow, are equal to the interior opposite angles Green, Red.

But the adjacent angles Green, Yellow, are equal to two right angles.

Then because the interior opposite angles Green, Red, and two right angles, are separately equal to the two adjacent angles Green, Yellow, they are equal to each other.

Diagram 44.

By means of Fig. A,—

1. Prove the interior opposite angles Green, Yellow, equal to two right angles, using the angle Red.

2. Prove the same, using the angle Blue.

3. Prove the same, using the anglee g b.

4. Prove the same, using the anglef h d.

5. Go through the same demonstrations again, naming the angles by their letters instead of by their colors.

6. Prove the interior opposite angles Red, Blue, equal to two right angles, using the angle Yellow.

7. Prove the same, using the angle Green.

8. Prove the same, using the anglee g a.

9. Prove the same, using the anglec h f.

10. Go through the same demonstrations again, calling the angles by their letters instead of by their colors.

By means of Fig. B,—

11. Prove the interior opposite anglesa g h,g h c, equal to two right angles, using the angleg h d.

12. Prove the same, using the anglec h f.

13. Prove the same, using the anglea g e.

14. Prove the interior opposite anglesb g h,g h d, equal to two right angles, using the anglea g h.

15. Prove the same, using the anglee g b.

16. Prove the same, using the anglef h d.

Compare the angles Yellow, Green, each with its exterior opposite angle, and see if you can prove that the exterior opposite anglese g b,f h d, are also equal to two right angles.

Diagram 45.

Suppose we do not know whether the linesa b,c d, are parallel, or not;

But, by measuring, we find that the interior angles Blue, Yellow, on the same side of the secant[4]linee f, are equal to two right angles:

4.“Secant” means “cutting.”

The adjacent angles Blue, Red, are equal to what?

Then, if the interior angles Blue, Yellow, are equal to two right angles,

And the adjacent angles Blue, Red, are also equal to two right angles,

What do you infer?

From the interior angles Blue, Yellow, take away the angle Blue: what remains?

From the adjacent angles Blue, Red, take away the same angle Blue: what remains?

What do you know of the two remainders?

The angle Red measures the direction of the lineg bfrom what line?

The equal angle Yellow measures the direction of the lineh dfrom what line?

Then if the linesg b,h d, have the same direction from the linee f, what do you call them?

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