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_.
But the new angle _b c d_ is equal to the angles _b c e_, _e c d_.
Then because the new angle _b c d_, and the angles _a_ and _b_ are
separately equal to the angles _b c e_, _e c d_, they are equal to
each other.
[Illustration]
PROPOSITION XI. THEOREM.
DEVELOPMENT LESSON.
Let the figure _a b c_ be a triangle.
Produce the side _a c_ to _d_.
By the last theorem, the angle _b c d_ is equal to what angles of the
triangle?
What angle must we add to these angles to make up the three angles of
the triangle?
If we add the same angle to the angle _b c d_, what adjacent angles do
we get?
Then the three angles of the triangle, _a_, _b_, and _c_, are equal to
what two angles?
But the adjacent angles _a c b_ and _b c d_ are equal to what?
Then, because the three angles of the triangle, _a_, _b_, and _c_, and
two right angles, are separately equal to the two adjacent angles
_c_ and _b c d_.
What new thing have you found out?
DEMONSTRATION.
We wish to prove that
_The three angles of any triangle are equal to two right angles._
Let the figure _a b c_ be a triangle; then will the sum of the angles
_a_, _b_, and _c_, be equal to two right angles.
For, produce the side _a c_ to _d_.
The new angle _b c d_ is equal to the sum of the angles _a_ and _b_.
If to the angles _a_ and _b_ we add the angle _c_, we shall have the
three angles of the triangle.
If to the angle _b c d_ we add the same angle _c_, we shall have the
adjacent angles _c_ and _b c d_.
Then the three angles of the triangle _a_, _b_, _c_, are equal to the
adjacent angles _c_ and _b c d_.
But the adjacent angles _c_ and _b c d_ are equal to two right angles.
Then, because the three angles of the triangle are equal to the
adjacent angles _c_ and _b c d_, they are equal to two right angles.
[Illustration]
PROPOSITION XII. THEOREM.
DEVELOPMENT LESSON.
Let the Fig. A B C D be a parallelogram.
Produce the side C D to F.
Because the straight line B D intersects the parallels A B and C F,
the angle B is equal to what other angle?
Because the straight line C F intersects the parallels A C and B D,
the angle C is equal to what other angle?
Then what follows from this?
To what angle did you find two others equal?
What two angles did you find equal to it?
What axiom do you think of?
See if you can go through the demonstration without reading it even
once.
DEMONSTRATION.
We wish to prove that
_The opposite angles of a parallelogram are equal to each other._
Let the Fig. A B C D be a parallelogram.
Then will any two opposite angles, as B and C, be equal to each other.
For produce the line C D to F.
Because the straight line B D meets the two parallels A B and C F,
The interior alternate angles B and E are equal to each other.
Because the straight line C F meets the two parallels B D and A C,
The opposite exterior and interior angles C and E are equal to each
other.
Then, because the angles B and C are separately equal to the angle E,
they are equal to each other.
* * * * *
1. Prove the same by producing the line A B towards the left.
2. Prove the same by producing the line B D downwards.
3. Prove the angles A and D equal to each other by producing the line
C D towards the left.
4. Prove the same by producing the line D B upwards.
5. See if you can prove the same by drawing a diagonal through the
points A and D.
[Illustration]
PROPOSITION XIII. THEOREM.
DEVELOPMENT LESSON.
In these two triangles we have tried to make the side _a b_ of the one
equal to the side _d e_ of the other; the side _a c_ of the one
equal to the side _d f_ of the other; and the included angle _b a c_
of the one equal to the included angle _e d f_ of the other.
We now wish to find out if the third side _b c_ of the one is equal to
the third side _e f_ of the other, and if the two remaining angles
_b_ and _c_ of the one are equal to the two remaining angles _e_ and
_f_ of the other.
Suppose we were to cut the triangle _d e f_ out of the page, and place
it upon the triangle _a b c_, so that the line _d e_ should fall
upon the line _a b_, and the point _d_ upon the point _a_.
As the line _d e_ is equal to the line _a b_, upon what point will the
point _e_ fall?
If the angle _e d f_ were less than the angle _b a c_, would the line
_d f_ fall within or without the triangle?
If the angle _e d f_ were greater than the angle _b a c_, where would
the line _d f_ fall?
Since the angle _a_ is equal to _d_, where, then, must the line _d f_
fall?
As the line _d f_ is equal to the line _a c_, upon what point will the
point _f_ fall?
Then, if the point _e_ falls upon the point _b_, and the point _f_
upon the point _c_, where will the line _e f_ fall?
Now, because the three sides of the triangle _d e f_ exactly fall upon
the three sides of the triangle _a b c_, we say _the two magnitudes
coincide throughout their whole extent_, and are therefore equal.
What three parts of the triangle _a b c_ did we suppose to be equal to
three corresponding parts of the triangle _d e f_ before we placed
one upon the other.
What line of the one do we _find_ equal to a line in the other?
What two angles of the one do we _find_ equal to two angles in the
other?
What do you think of the areas of the triangles?
[Illustration]
DEMONSTRATION.
We wish to prove, that,
_If two triangles have two sides, and the included angle of the one
equal to two sides and the included angle of the other, each to
each, the two triangles are equal in all respects._
Let the triangles _a b c_ and _d e f_ have the side _a b_ of the one
equal to the side _d e_ of the other; the side _a c_ of the one
equal to the side _d f_ of the other; and the included angle _b a c_
of the one equal to the included angle _e d f_ of the other, each to
each; then will the two triangles be equal in all their parts.
For, place the triangle _d e f_ upon the triangle _a b c_, so that the
line _d e_ shall fall upon the line _a b_, with the point _d_ upon
the point _a_.
Because the line _d e_ is equal to the line _a b_, the point _e_ will
fall upon the point _b_.
Because the angle _e d f_ is equal to the angle _b a c_, the line _d
f_ will fall upon the line _a c_.
Because the line _d f_ is equal to the line _a c_, the point _f_ will
fall upon the point _c_.
Then, because the point _e_ is on the point _b_, and the point _f_ on
the point _c_, the line _e f_ will coincide with the line _b c_, and
the two triangles will be found equal in all their parts;
That is, the angle _e_ is found to be equal to the angle _b_, the
angle _f_ to the angle _c_, the line _e f_ to the line _b c_, and
the area of the triangle _a b c_ to the area of the triangle _d e
f_.
[Illustration]
PROPOSITION XIV. THEOREM.
DEVELOPMENT LESSON.
In these two triangles we have tried to make the angle _b_ of the one
equal to the angle _e_ of the other; the angle _c_ of the one equal
to the angle _f_ of the other; and the included side _b c_ of the
one equal to the included side _e f_ of the other.
We now wish to find out if the remaining angle _a_ of the one is equal
to the remaining angle _d_ of the other, and if the two remaining
sides _a b_ and _a c_ of the one are equal to the two remaining
sides _d e_ and _d f_ of the other.
Suppose we were to cut the triangle _d e f_ out of the page and place
it upon the triangle _a b c_, so that the line _e f_ shall fall upon
the line _b c_, with the point _e_ upon the point _b_.
Because the line _e f_ is equal to the line _b c_, upon what point
will the point _f_ fall?
Because the angle _e_ is equal to the angle _b_, where will the line
_e d_ fall?
Because the angle _f_ is equal to the angle _c_, where will the line
_d f_ fall?
Then, if the line _d e_ falls upon the line _a b_ and the line _d f_
upon the line _a c_, where will the point _d_ fall?
Now because the three sides of the triangle _d e f_ exactly fall upon
the three sides of the triangle _a b c_, we say _the two magnitudes
coincide throughout their whole extent, and are therefore equal_.
Suppose the angle _e_ were greater than the angle _b_, would the line
_e d_ fall within or without the triangle?
If it were less, where would the line fall?
Why does the line _d e_ fall exactly upon the line _a b_?
[Illustration]
DEMONSTRATION.
We wish to prove that,
_If two triangles have two angles, and the included side of the one
equal to two angles and the included side of the other, each to
each, the two triangles are equal to each other in all respects._
Let the triangles _a b c_ and _d e f_ have the angle _b_ of the one
equal to the angle _e_ of the other; the angle _c_ of the one equal
to the angle _f_ of the other; and the included side _b c_ of the
one equal to the included side _e f_ of the other, each to each;
then will the two triangles be equal in all their parts.
For place the triangle _d e f_ upon the triangle _a b c_, so that the
line _e f_ shall fall upon the line _b c_, with the point _e_ upon
the point _b_.
Because the line _e f_ is equal to the line _b c_ the point _f_ will
fall upon the point _c_.
Because the angle _e_ is equal to the angle _b_, the line _e d_ will
fall upon the line _b a_, and the point _d_ will be somewhere in the
line _b a_.
Because the angle _f_ is equal to the angle _c_, the line _f d_ will
fall upon the line _c a_, and the point _d_ will be somewhere in the
line _c a_.
Then, because the point _d_ is in the two lines, _b a_ and _c a_, it
must be in their intersection, or upon the point _a_.
And, as the two triangles coincide throughout their whole extent, they
are equal in all their parts.
That is, the angle _a_ is found to be equal to the angle _d_; the side
_b a_ to the side _e d_; the side _c a_ to the side _f d_; and the
area of the triangle _a b c_ to the area of the triangle _d e f_.
[Illustration]
PROPOSITION XV. THEOREM.
DEMONSTRATION.
We wish to prove that
_The opposite sides of any parallelogram are equal._
Let the figure _a b c d_ be a parallelogram; then will the sides _a b_
and _c d_ be equal to each other; likewise the sides _a d_ and _b
c_.
For, draw the diagonal _b d_.
Because the figure is a parallelogram, the sides _a b_ and _d c_ are
parallel, and the interior alternate angles _n_ and _o_ are equal.
Because the figure is a parallelogram, the interior alternate angles
_r_ and _m_ are equal.
Then the two triangles _a d b_, _b d c_, have two angles and the
included side of the one equal to two angles and the included side
of the other, each to each, and are therefore equal;
And the side _a b_ opposite the angle _m_ is equal to the side _c d_
opposite the equal angle _r_;
And the side _a d_ opposite the angle _n_ is equal to the side _b c_
opposite the equal angle _o_.
TEST.
Prove the same by drawing a diagonal from _a_ to _c_.
[Illustration]
PROPOSITION XVI. THEOREM.
DEVELOPMENT LESSON.
Suppose A B to be a straight line, and C any point out of it.
From the point C draw a perpendicular C F to A B.
Let us see if this perpendicular is not shorter than any other line we
can draw from the same point to the same line.
Draw any other line from C to A B as C E.
Now, as C E is any line whatever other than a perpendicular, if we
find that the perpendicular C F is shorter than it we must conclude
that it is the shortest line that can be drawn from C to A B.
Produce C F until F D is equal to C F, and then join E and D.
In the triangles E F C, E F D, what two sides were drawn equal?
What line is a side to each?
How great an angle is C F E?
What is a right angle?
Then how do the angles C F E and E F D compare with each other?
If the two triangles E F C, E F D, have the side C F of the one equal
to the side F D of the other, the side E F common to both, and the
included angle E F C of the one equal to the included angle E F D of
the other, each to each, what do you infer?
Then what third side of the one have you found equal to a third side
of the other?
C E is what part of the broken line C E D?
C F is what part of the line C D?
Which is shorter, the straight line C D, or the broken line C E D?
Then how does the half of C D or C F compare with the half of C E D or
C E?
If C E is any line whatever other than a perpendicular, what may we
now say of the perpendicular from the point C to the straight line A
B?
[Illustration]
DEMONSTRATION.
We wish to prove that
_A perpendicular is the shortest distance from a point to a straight
line._
Let A B be a straight line, and C A point out of it; then will the
perpendicular C E be the shortest line that can be drawn from the
point to the line.
For draw any other line from C to A B, as C F.
Produce C E until E D equals C E, and join F D.
The two triangles F E C, F E D, have the side C E of the one equal to
the side E D of the other, the side F E common, and the included
angle F E C of the one equal to the included angle F E D of the
other, they are therefore equal, and the side C F equals the side F
D.
But the straight line C D is the shortest distance between the two
points C D; therefore it is shorter than the broken line C F D.
Then C E, the half of C D, is shorter than C F, the half C F D.
And, as C F is any line other than a perpendicular, the perpendicular
C E is the shortest line that can be drawn from C to A B.
[Illustration]
PROPOSITION XVII. THEOREM.
DEMONSTRATION.
We wish to prove that
_A tangent to a circumference is perpendicular to a radius at the
point of contact._
Let the straight line A B be tangent at the point D to the
circumference of the circle whose centre is C.
Join the centre C with the point of contact D, the tangent will be
perpendicular to the radius C D.
For draw any other line from the centre to the tangent, as C F.
As the point D is the only one in which the tangent touches the
circumference, any other point, as F, must be without the
circumference.
Then the line C F, reaching _beyond_ the circumference, must be longer
than the radius C D, which would reach only to it; therefore C D is
shorter than any other line which can be drawn from the point C to
the straight line A B; therefore it is perpendicular to it.
PROPOSITION XVIII. THEOREM.
DEMONSTRATION.
[Illustration]
We wish to prove that
_In any isosceles triangle, the angles opposite the equal sides are
equal._
Let the triangle A B C be isosceles, having the side A B equal to the
side A C; then will the angle B, opposite the side A C, be equal to
the angle C, opposite the equal side A B.
For draw the line A D so as to divide the angle A into two equal
parts, and let it be long enough to divide the side B C at some
point as D.
Now the two triangles A D B, A D C, have the side A B of the one equal
to the side A C of the other, the side A D common to both, and the
included angle B A D of the one equal to the included angle C A D of
the other; therefore the two triangles are equal in all respects,
and the angle B, opposite the side A C, is equal to the angle C,
opposite the side A B.
[Illustration]
PROPOSITION XIX. THEOREM.
DEMONSTRATION.
We wish to prove that,
_If two triangles have the three sides of the one equal to the three
sides of the other, each to each, they are equal in all their
parts._
Let the two triangles A B C, A D C, have the side A B of the one equal
to the side A D of the other; the side B C of the one equal to the
side D C of the other, and the third side likewise equal; then will
the two triangles be equal in all their parts.
For place the two triangles together by their longest side, and join
the opposite vertices B and D by a straight line.
Because the side A B is equal to the side A D, the triangle B A D is
isosceles, and the angles A B D, A D B, opposite the equal sides are
equal.