Question

Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof? overline HK is a perpendicular bisector of overline IJ , and L is the midpoint of overline IJ. M is a point on the perpendicular bisector, overline HK By the definition of a perpendicular bisector, I know that overline IM≌ overline JM. By the definition of a perpendicular bisector, I also know that ∠ MLI and ∠ MLJ are right angles. ∠ MLI≌ ∠ MLJ because of the Right Angle Congruence Theorem. I can also say that overline ML≌ overline ML by the Reflexive Property of Congruence. With this information, I know that △ MLI≌ △ MLJ by the SAS Congruence Theorem. Since the triangles are congruent, the CPCTC Theorem allows me to know that overline IL≌ overline JL. Knowing that these segments are congruent proves the Perpendicular Bisector Theorem. (1 point) The SAS Congruence Theorem tells you that △ MLI≌ △ JLM , not that △ MLI≌ △ MLI The definition of a perpendicular bisector tells you that ∠ LMI and ∠ LMJ are right angles, not that ∠ MLI and ∠ MLJ are right angles. The definition of a perpendicular bisector tells you that overline IL≌ overline JL , not that overline IM≌ overline JM. Lenny did not make a mistake. His proof is correct.

Answer 1

He used the wrong angle for SAS. In triangles MLI and MLJ he claimed ML ≅ ML and IM ≅ JM and ∠MLI ≅ ∠MLJ. But ∠MLI is the angle between ML and LI, not between ML and IM, so that angle is not included between the two sides ML and IM. SAS requires the congruent angle to be included between the two congruent sides, so SAS does not apply here.