Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/60

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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.