isosceles, and the angles C B D, C D B, opposite the equal sides are
equal.
If to the angle A B D we add the angle D B C, we shall have the angle
A B C.
And if to the equal of A B D, that is, A B D, we add the equal of D B
C, that is, B D C, we shall have the angle A D C.
Therefore the angle A B C is equal to the angle A D C.
Then the two triangles A B C, A D C, have two sides, and the included
angle of the one equal to two sides and the included angle of the
other, each to each, and are equal in all their parts; that is, the
three angles of the one are equal to the three angles of the other,
and their areas are equal.
[Illustration]
PROPOSITION XX. THEOREM.
DEMONSTRATION.
We wish to prove that
_An angle at the circumference is measured by half the arc on which
it stands._
Let B A D be an angle whose vertex is in the circumference of the
circle whose centre is C; then will it be measured by half the arc B
D.
For through the centre draw the diameter A E, and join the points C
and B.
The exterior angle E C B is equal to the sum of the angles B and B A
C.
Because the sides C A, C B, are radii of the circle, they are equal,
the triangle is isosceles, the angles B and B A C opposite the equal
sides are equal, and the angle B A C is half of both.
Then, because the angle B A C is half of B and B A C, it must be half
of their equal E C B.
But E C B, being at the centre, is measured by B E; then half of it,
or B A C, must be measured by half B E.
In like manner, it may be proved that the angle C A D is measured by
half E D.
Then, because B A C is measured by half B E, and C A D by half E D,
the whole angle B A D must be measured by half the whole arc B D.
SECOND CASE.
Suppose the angle were wholly on one side of the centre, as F A B.
Draw the diameter A E and the radius B C as before.
Prove that the angle B A E is measured by half the arc B E.
Draw another radius from C to F, and prove that F A E is measured by
half the arc F E.
Then, because the angle F A E is measured by half the arc F E, and the
angle B A E is measured by half the arc B E,
The difference of the angles, or F A B, must be measured by half the
difference of the arcs, or half of F B.
[Illustration]
PROPOSITION XXI. THEOREM.
DEMONSTRATION.
We wish to prove that
_Parallel chords intercept equal arcs of the circumference._
Let the chords A B, C D, be parallel; then will the intercepted arcs A
C and B D be equal.
For draw the straight line B C.
Because the lines A B and C D are parallel, the interior alternate
angles A B C, B C D, are equal.
But the angle A B C is measured by half the arc A C;
And the angle B C D is measured by half the arc B D:
Then, because the angles are equal, the half arcs which measure them
must be equal, and the whole arcs themselves must be equal.
[Illustration]
PROPOSITION XXII. THEOREM.
DEMONSTRATION.
We wish to prove that
_The angle formed by a tangent and a chord meeting at the point of
contact is measured by half the intercepted arc._
Let the tangent C A B and the chord A D meet at the point of contact
A; then will the angle B A D be measured by half the intercepted arc
A D.
For draw the diameter A E F.
Because A B is a tangent, and A E a radius at the point of contact,
the angle B A F is a right angle, and is measured by the semicircle
A D F.
Because the angle F A D is at the circumference, it is measured by
half the arc D F.
Then the difference between the angles B A F and D A F, or B A D, must
be measured by half the difference of the arcs A D F and D F, or A
D;
That is, the angle B A D is measured by half the arc A D.
[Illustration]
PROPOSITION XXIII. THEOREM.
DEMONSTRATION.
We wish to prove that
_A tangent and chord parallel to it intercept equal arcs of the
circumference._
Let A B be tangent to the circumference at the point D, and let C F be
a chord parallel to the tangent; then will the intercepted arcs C D
and D F be equal.
For from the point of contact D, draw the straight line D C.
Because the tangent and chord are parallel, the interior alternate
angles A D C and D C F are equal.
But the angle A D C, being formed by the tangent D A and the chord D
C, is measured by half the intercepted arc D C;
And the angle D C F, being at the circumference, is measured by half
the arc on which it stands, D F:
Then, because the angles are equal, the half arcs which measure them
are equal, and the arcs themselves are equal.
[Illustration]
PROPOSITION XXIV. THEOREM.
DEMONSTRATION.
We wish to prove that
_The angle formed by the intersection of two chords in a circle is
measured by half the sum of the intercepted arcs._
Let the chords A B and C D intersect each other in the point E; then
will the angle B E D or A E C be measured by half the sum of the
arcs A C, B D.
For from the point C draw C F parallel to A B.
Because the chords A B and C F are parallel, the arcs A C, B F, are
equal.
Add each of these equals to B D, and we have B D plus A C equal to B D
plus B F; that is, the sum of the arcs B D, A C, is equal to the arc
F D.
Because the chords A B, C F, are parallel, the opposite exterior and
interior angles D E B, D C F, are equal.
But D C F is an angle at the circumference, and is therefore measured
by half the arc F D.
Then the equal angle D E B must be measured by half of the arc F D, or
its equal B D, plus A C.
[Illustration]
PROPOSITION XXV. THEOREM.
DEMONSTRATION.
We wish to prove that
_The angle formed by two secants meeting without a circle is measured
by half the difference of the intercepted arcs._
Let the secants A B, A C, intersect the circumference in the points D
and E; then will the angle B A C be measured by half the difference
between the arcs B C and D E.
For from the point D draw the chord D F parallel to E C.
Because A C and D F are parallel, the opposite exterior and interior
angles B D F and B A C are equal.
Because the chords D F, E C, are parallel, the arcs D E and F C are
equal.
If from the arc B C we take the arc D E, or its equal F C, we shall
have left the arc B F;
But the angle B D F, being at the circumference, is measured by half
the arc B F:
Then the equal of B D F, or B A C, must be measured by half the arc B
F, or half the difference between the intercepted arcs B C and D E.
APPENDIX.
NOTE A.—To those teachers who think that the line should be derived
from a surface, and the surface from a solid, the author would say,
that, according to his experience, children apprehend the ideas
conveyed by the terms _line_ and _surface_ as readily as they do any
ideas whatever; and that, therefore, there seems to be no necessity
for extraordinary care in this case to avoid giving wrong
impressions.
Still, if it be considered desirable in this manner to derive lines
and surfaces, it will be apparent that all that can be done in the
matter is to give such instruction only by way of a preliminary
lesson.
* * * * *
NOTE B.—Crooked and curved lines are here treated of before straight
lines, because the first two are defined by means of an affirmative
property,—they _do_ change direction; while the last is defined by
means of the absence of one,—they do _not_ change direction. It is
easier for a child to comprehend what is than what is _not_.
* * * * *
NOTE C.—If the pupils are old enough, they may be shown that
vertical lines cannot be parallel, but only seem so on account of
their shortness and nearness to each other.
* * * * *
NOTE D.—This definition may be considered objectionable because
_rhomboid_ means like a rhomb. That the more general figure, the
rhomboid, has been named from the more restricted one, the rhomb, is
unfortunate, because it interferes with the symmetry of the
nomenclature. The rhomb possesses all the properties of the
rhomboid, and should, therefore, when these are considered, be
called by the same name; its additional property entitles it to a
name which should comprehend the other names. If the rectangle had
been called a squaroid, the difficulty would have been repeated.
* * * * *
NOTE E.—If teachers consider it desirable, they may require the
class to prove, by way of corollary, such propositions as assert the
parallelism of the lines when the interior alternate angles are
equal, when the opposite exterior and interior angles are equal,
&c., in continuation of what has already been done.
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End of Project Gutenberg's Marks' first lessons in geometry, by Bernhard Marks