Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/59

Rh Because the straight line AD which meets the two straight lines BC, EF, makes the alternate angles BAD, ADC equal to one another, [Construction. EF is parallel to BC. [I. 27.

Wherefore the straight line EAF is drawn through the given point A, parallel to the given straight line BC.

PROPOSITION 32. THEOREM.

If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

Let ABC be a triangle, and let one of its sides BC be produced to D : the exterior angle ACD shall be equal to the two interior and opposite angles CAB,ABC; and the three interior angles of the triangle, namely, ABC, BCA, CAB shall be equal to two right angles.

Through the point C draw CE parallel to AB. [1.31.

Then, because AB is par- allel to CE, and AC falls on them, the alternate angles BAC, ACE are equal. [I. 29. Again, because AB is parallel to CE, and BD falls on them, the exterior angle ECD is equal to the interior and opposite angle ABC. [I. 29. But the angle ACE was shewn to be equal to the angle BAC; therefore the whole exterior angle ACD is equal to the two interior and opposite angles CAB, ABC. [Axiom 2.

To each of these equals add the angle ACB; therefore the angles ACD, ACB are equal to the three angles CBA, BAC, ACB. [Axiom 2. But the angles ACD, ACB are together equal to two right angles ; [I. 13. Therefore also the angles CBA, BAC, ACB are together equal to two right angles. [Axiom 1.

Wherefore, if a side of any triangle &c.