Question

proofs about angles quick check
lenny wrote a paragraph proof of the perpendicular bisector theorem. what mistake did lenny make in his proof?
hk is a perpendicular bisector of ij, and l is the mid - point of ij. m is a point on the perpendicular bisector, hk. by the definition of a perpendicular bisector, i know that im ≅ 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 ml ≅ 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 il ≅ jl. knowing that these segments are congruent proves the perpendicular bisector theorem.
(1 point)
the definition of a perpendicular bisector tells you that il ≅ jl, not that im ≅ jm
the sas congruence theorem tells you that △mli ≅ △jlm, not that △mli ≅ △mlj
the definition of a perpendicular bisector tells you that ∠lmi and ∠lmj are right angles, not that ∠mli and ∠mlj are right angles.
lenny did not make a mistake. his proof is correct.

Explanation:

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. The definition of a perpendicular bisector gives that any point on it (like $M$) is equidistant from the endpoints of the bisected segment ($I$ and $J$), so $IM\cong JM$. Lenny's proof is correct as he correctly uses definitions, congruence theorems and CPCTC.

Answer:

Lenny did not make a mistake. His proof is correct.