Exercise 32 Page 298

Since PM ⊥ AB, we can say that ∠ PMA and ∠ PMB are right angles by definition of perpendicular lines. Definition of Perpendicular Lines m∠ PMA=m∠ PMB=90^(∘) Recall that all right angles are congruent. Definition of Angle Congruence ∠ PMA≅∠ PMB Given that PM bisects AB, the distance from A to M is the same as the distance from M to B. Definition of Bisector AM=BM If the line segments have the same lengths, then they are congruent. Definition of Segment Congruence AM≅BM Considering the Reflexive Property of Congruence, we can write the next step of our proof. Reflexive Property of Congruence PM≅PM So far, we have shown that two sides and the included angle of △ PMA are congruent to two sides and the included angle of △ PMB. Therefore, by the Side-Angle-Side Congruence Theorem, these two triangles are congruent. SAS Congruence Theorem △ PMA≅△ PMB Knowing that the corresponding parts of congruent triangles are congruent, we can say that AP is congruent to BP. Definition of Congruent Triangles AP ≅ BP Finally, to complete our proof, we will use the fact that two congruent segments have the same length. Definition of Segment Congruence AP=BP Combining all of these steps, we can form our two-column proof.