Tangents to a Circle are Congruent

Prove that tangents to a circle from a point are congruent.

Circle O is given. Point P is outside of circle O. In accordance with Definition 6.3, construct segments PA and PB so they form tangents to circle O. Also construct radii AO and BO. Two triangles are now formed – triangle PAB and triangle OAB. As radii are congruent (Theorem 6.1) then AO and BO are congruent. The two radii intersect the two tangent lines perpendicularly (Theorem 6.5). Segment PO then becomes the hypotenuse for triangles PAO and PBO. Segments are also congruent to themselves (AC02) so in both triangles PO is congruent. According to Theorem 4.78, triangles PAO and PBO are congruent as they have congruent hypotenuses and congruent corresponding legs. Theorem 4.20 says that all sides of congruent triangles are congruent; therefore, it can be concluded that PA nd PB are congruent as they are corresponding sides.