36 | EUCLID'S ELEMENTS. |
Corollary 1. All the interior angles of any recti-lineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
For any rectilineal figure ABCDE can be divided into as many triangles as the figure has sides, by drawing; straight lines from a point F within the figure to each of its angles.
And by the preceding proposition,
all the angles of these triangles are
equal to twice as many right angles
as there are triangles, that is, as the
figure has sides.
And the same angles are equal to the
interior angles of the figure, together
with the angles at the point F, which
is the common vertex of the triangles,
that is, together with four right angles. [I. 15. Corollary 2.
Therefore all the interior angles of the figure, together with
four right angles, are equal to twice as many right angles
as the figure has sides.
Corollary 2. All the exterior angles of any recti-lineal figure are together equal to four right angles.
Because every interior angle
ABC, with its adjacent exterior
angle ABD, is equal to two
right angles ; [I. 13.
therefore all the interior angles
of the figure, together with all
its exterior angles, are equal to
twice as many right angles as
the figure has sides.
But, by the foregoing Corollary all the interior angles of the
figure, together with four right angles, are equal to twice
as many right angles as the figure has sides.
Therefore all the interior angles of the figure, together with
all its exterior angles, are equal to all the interior angles of
the figure, together with four right angles.
Therefore all the exterior angles are equal to four right;
angles.