Bernhard Marks.

Marks' first lessons in geometry, objectively presented online

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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?

[Illustration: Diagram 31.]


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.

[Illustration: Diagram 32.]


TEST LESSON.

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.

[Illustration: Diagram 33.]


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._

[Illustration: Diagram 34.]


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]

Footnote 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.


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?

[Illustration: Diagram 35.]


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.

[Illustration: Diagram 36.]


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_.

[Illustration: Diagram 37.]


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?


DEMONSTRATION.

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.

[Illustration: Diagram 38.]


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.

[Illustration: Diagram 39.]


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?


DEMONSTRATION.

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.

[Illustration: Diagram 40.]


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_.

[Illustration: Diagram 41.]


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?


DEMONSTRATION.

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.

[Illustration: Diagram 42.]


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_.

[Illustration: Diagram 43.]


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.

[Illustration: Diagram 44.]


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.

[Illustration: Diagram 45.]


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:

Footnote 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 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?

[Illustration: Diagram 46.]


DEMONSTRATION.

We wish to prove, that,

_If a straight line intersects two other straight lines so that two
interior angles on the same side of the intersecting line are equal
to two right angles, the two lines are parallel._

Let the straight line _e f_ intersect the two straight lines _a b_, _c
d_, in the points _g_ and _h_, so that the angles Red, Blue, are
equal to two right angles.

Then will the lines _a b_, _c d_, be parallel.

For the angles Red, Blue, are supposed equal to two right angles.

The adjacent angles Red, Green, are known to be also equal to two
right angles.

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

If from the interior angles Red, Blue, we take away the angle Red, we
have left the angle Blue.

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

Then the angle Blue is equal to the angle Green.

But the angle Blue measures the direction of the line _h d_ from the
line _e f_.

And the angle Green measures the direction of the line _g b_ from the
line _e f_.

Then the lines _g b_, _h d_, have the same direction, and are
parallel.


TEST LESSON.

1. Prove the same without the colors.

2. Prove the same, using the angle _f h d_.

3. Prove the same, supposing the angles _a g h_, _g h c_, equal to two
right angles, and using the angle _a g e_.

4. Prove the same, using the angle _c h f_.

See Note E, Appendix.


PROPOSITION VIII. THEOREM.

The following demonstration is very easy. Read it once, and see if you
can go through it without a second reading:—


DEMONSTRATION.

[Illustration]

We wish to prove that

_The sum of any two sides of a triangle is greater than the third
side._

Let the figure _a b c_ be a triangle, then will the sum of any two
sides, as _a c_, _c b_, be greater than the third side _a b_.

For the straight line _a b_ is the shortest distance between the two
points _a_ and _b_, and is therefore less than the broken line _a c
b_.


PROPOSITION IX. PROBLEM.

The following solution is so easy that you will understand it at
once:—

We wish

_To construct an equilateral triangle on a given straight line._

[Illustration]


SOLUTION.

Let _a b_ be the given line.

With the point _a_ as a centre, and _a b_ as a radius, draw the
circumference of the circle, or a part of one.

With the point _b_ as a centre, and the same radius _a b_, draw
another circumference, or a part of one.

From the point _c_, in which the circumferences or arcs intersect,
draw the straight lines _a c_ and _b c_.

Now, because the lines _a b_ and _a c_ are radii of the same circle,
they are equal.

And, because the lines _a b_ and _b c_ are radii of the same circle,
they are also equal.

Then, because the two lines _a c_, _b c_, are separately equal to the
line _a b_, they are equal to each other, and the triangle is
equilateral.

[Illustration]


PROPOSITION X. THEOREM.


DEVELOPMENT LESSON.

Let the figure _a b c_ be a triangle.

Produce the side _a c_ to _d_.

We have now another angle, _b c d_, and we wish to find out if it is
equal to any of the angles of the triangle.

From the point _c_ draw the line _c e_ parallel to _a b_.

Because the straight line _a d_ intersects the two parallels _a b_, _c
e_, the angle _a_ is equal to what other angle?

Because the straight line _b c_ intersects the two parallels _a b_, _c
e_, the angle _b_ is equal to what other angle?

Then the angles _a_ and _b_ are equal to what two angles?

How does the angle _b c d_ compare with the angles _b c e_, _e c d_?

Then, if the angles _a_ and _b_, on the one hand, and the angle _b c
d_, on the other, are separately equal to the angles _b c e_, _e c
d_,

What have you found out?

What axiom have you just employed?

To what same thing have you found two other things equal?

What two things did you find equal to it?


DEMONSTRATION.

We wish to prove, that,

_If any side of a triangle be produced, the new angle formed will be
equal to the sum of the angles that are not adjacent to it._

Let _a b c_ be a triangle.

Produce the side _a c_ to _d_; then will the new angle _b c d_ be
equal to the sum of the angles _a_ and _b_.

For from the point _c_ draw _c e_ parallel to _a b_.

Then, because the straight line _a d_ intersects the two parallels _a
b_, _c e_, in the points _a_ and _c_,

The opposite exterior and interior angles _a_ and _e c d_ are equal to
each other.

And because the straight line _b c_ intersects the same parallels in
the points _b_ and _c_,

The interior alternate angles _b_ and _b c e_ are equal.

Then the angles _a_ and _b_ of the triangle are equal to the angles _b
c e_ and _e c d_.


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Online LibraryBernhard MarksMarks' first lessons in geometry, objectively presented → online text (page 4 of 6)