Marks' first lessons in geometry / In two parts. Objectively presented, and designed for the use of primary classes in grammar schools, academies, etc.

AXIOMS ILLUSTRATED.

Axiom 1.

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.

Axiom 2.

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.

Axiom 3.

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.

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.

Axiom 5.

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

Axiom 6.

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

Axiom 7.

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

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

Axiom 9.

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

Axiom 10.

A straight line is the shortest distance between two points.

Axiom 11.

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

THEOREMS ILLUSTRATED.

DEVELOPMENT LESSON.

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

Note.—The word space, as here used, means angular 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.

PROPOSITION I. THEOREM.

DEVELOPMENT LESSON.

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

Place a paper square corner or right angle on the line a b at the left of c d with its vertex at c.

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

Now place another square corner on the line a b to the right of the line c d, and with its vertex at the point c.

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

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

Then if the two adjacent angles Green, Red, are equal to all the angular space on the line a b, and the two right angles are also equal to the same space, what do you infer concerning the adjacent angles and the two right angles?

What axiom do you apply when you say that the adjacent angles are equal to the two right angles?

To what same thing did 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 you find true?

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 line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be?

See now if you can understand the following demonstration:—

DEMONSTRATION.

We wish to prove that

Any two adjacent angles are equal to two right angles.

Let the two straight lines a b, m n, intersect each other in the point c. (Diagram 30.)

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

For, from the point c, draw the straight line c d so as to make the angles a c d, d c b, right angles.

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

The right angles a c d, d c b, are also equal to all the angular space on the line a b.

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

TEST QUESTIONS.

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?

TEST LESSON.

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.

By means of Fig. A prove,—

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

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

By means of Fig. B prove,—

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

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

By means of Fig. C prove,—

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

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

By means of Fig. D prove,—

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

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

PROPOSITION II. THEOREM.

DEVELOPMENT LESSON.

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 lines a b and m n were 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.

DEMONSTRATION.

We wish to prove that

All vertical angles are equal to each other.

Let the straight lines a b, m n, intersect each other at the point c, 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]

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.

TEST QUESTIONS.

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

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?

OTHER METHODS OF DEMONSTRATION.

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.

TEST LESSON.

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 n equals m c b, by means of a c m.

10. a c n equals m c b, by means of b c n.

11. a c m equals n c b, by means of a c n.

12. a c m equals n c b, by means of m c b.

By means of Fig. D, prove that

13. m c a equals b c n, by means of a c n.

14. m c a equals b c n, by means of m c b.

15. m c b equals a c n, by means of m c a.

16. m c b equals a c n, by means of b c n.

PROPOSITION III. THEOREM.

DEVELOPMENT LESSON.

In the above diagram, the lines a b, c d, are parallel, and are intersected by the line e f at the points m and n.

The angle Red measures the difference of direction between the line m b and what other line?

The angle Yellow measures the difference of direction between the line n d and what other line?

Then, as the lines m b and n d are parallel, must there not be the same difference of direction between them and the line e 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?

We wish to prove that

Opposite exterior and interior angles are equal to each other.

Let the straight line e f intersect the two parallel straight lines a b, c d, at the points m and n.

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 lines m b and e f.

And the angle Yellow measures the difference of direction of the lines n d and e f.

But because the lines m 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.

TEST LESSON.

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 angles c n e, a m n, equal.

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

By means of Fig. B,—

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

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

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

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

PROPOSITION IV. THEOREM.

DEVELOPMENT LESSON.

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?

We wish to prove that

Any two interior alternate angles are equal to each other.

Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.

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.

TEST LESSON.

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 angles c n m, n m b, equal, using the angle f n d.

8. Prove the same, using the angle a m e.

9. Prove the interior alternate angles a m n, m n d, equal, using the angle e m b.

10. Prove the same, using the angle c n f.

PROPOSITION V. THEOREM.

DEVELOPMENT LESSON.

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 you find to be true?

We wish to prove that

Any two exterior alternate angles are equal to each other.

Let the straight line e f intersect the two parallel straight lines a b, c d, at the points m and n.

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.

TEST LESSON.

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 angles e m b, c n f, equal, using the angle a m n.

5. Prove the same, using the angle m n d.

By means of Fig. B,—

6. Prove that the exterior alternate angles c m e, f n b, are equal, using the angle n m d.

7. Prove the same, using the angle a n m.

8. Prove the exterior alternate angles e m d, a n f, equal, using the angle c m n.

9. Prove the same, using the angle m n b.

PROPOSITION VI. THEOREM.

DEVELOPMENT LESSON.

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?

DEMONSTRATION.

We wish to prove that

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

Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.

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.

TEST LESSON.

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 angle e g b.

4. Prove the same, using the angle f 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 angle e g a.

9. Prove the same, using the angle c 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 angles a g h, g h c, equal to two right angles, using the angle g h d.

12. Prove the same, using the angle c h f.

13. Prove the same, using the angle a g e.

14. Prove the interior opposite angles b g h, g h d, equal to two right angles, using the angle a g h.

15. Prove the same, using the angle e g b.

16. Prove the same, using the angle f h d.

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

PROPOSITION VII. THEOREM.

DEVELOPMENT LESSON.

Suppose we do not know whether the lines a 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] line e f, are equal to two right angles:

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 line g b from what line?

The equal angle Yellow measures the direction of the line h d from what line?

Then if the lines g b, h d, have the same direction from the line e f, what do you call them?