Nina has prepared the following two-column proof below. She is given that ∠OLN ≅ ∠LNO and she is trying to prove that OL ≅ ON. S
tepStatementReason1∠OLN ≅ ∠LNOGiven2Draw OE as a perpendicular bisector to LNby Construction3∠LEO ≅ ∠NEOTransitive Property of Equality4m∠LEO = 90°Definition of a Perpendicular Bisector5m∠NEO = 90°Definition of a Perpendicular Bisector6LE ≅ ENDefinition of a Perpendicular Bisector7ΔOLE ≅ ΔONESide-Angle-Side (SAS) Postulate8OL ≅ ONCPCTC
Step number 3 should really be step number 7, it should be placed after step 4, 5, and 6. The reason is because we won't know that ∠LEO ≅ ∠NEO until after we learn that LE ≅ EN. Because of this, step 3 is in the wrong spot (mistake number one). The secod mistake is that step 7, triangle OLE ≅ triangle ONE is congruent by Angle-Side-Angle (ASA) Postulate, not Side-Angle-Side (SAS) Postulate. It is congruent by ASA because we know that both triangles have equal angles N and L. We also know that the perpendicular bisector creates a 90° angle. So m∠LEO = 90° and ∠NEO = 90°. Therefore, we already have 2 congruent angles in both of the triangles. We also learn that line LE ≅ EN based on the definition of a perpendicular bisector, so we have know one that one side of each triangle is congruent. It is ASA and not AAS, because the ASA Postulate states that two angles and one included side of one triangle are congruent to two angles and one included side of another triangle.
