Proposition 32.
In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.
Let
ABC be a triangle, and let one side of it
BC be produced to
D;
I say that the exterior angle
ACD is equal to the two interior and opposite angles
CAB,
ABC, and the three interior angles of the triangle
ABC,
BCA,
CAB are equal to two right angles.
For let
CE be drawn through the point
C parallel to the straight line
AB. [
I. 31]
Then, since
AB is parallel to
CE,
and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [I. 29]
Again, since
AB is parallel to
CE,
and the straight line BD has fallen upon them, the exterior angle
ECD is equal to the interior and opposite angle
ABC. [
I. 29]
But the angle
ACE was also proved equal to the angle
BAC;
therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC.
Let the angle
ACB be added to each;
therefore the angles ACD, ACB are equal to the three angles ABC, BCA, CAB.
But the angles
ACD,
ACB are equal to two right angles; [
I. 13]
therefore the angles ABC, BCA, CAB are also equal to two right angles.
Therefore etc.
Q. E. D.
