flow proof

Question

given: om bisects <lmn lm≅

 nm 

prove : Δmol ≅ Δmon

Answer 1

A flow proof just shows how each given fact leads to the conclusion, so the goal here is to collect three pairs of corresponding parts and then name the congruence rule that uses them.

Start with what you were handed. You are given that LM ≅ NM, so those two sides are a matching pair right away. You are also given that MO bisects ∠LMN, and the definition of an angle bisector says it splits the angle into two congruent angles, so ∠LMO ≅ ∠NMO. That is your pair of angles.

The third pair is the one students usually miss, because nobody gives it to you. Look at the two triangles, ▵MOL and ▵MON, and notice that they share the side MO. Any segment is congruent to itself, so MO ≅ MO by the Reflexive Property. That is your third pair, and it is free.

Now check how those three pairs are arranged. You have side LM, then the angle at M, then side MO — and the angle sits between the two sides in each triangle. Side, included angle, side. That is SAS, so ▵MOL ≅ ▵MON by SAS.

Laid out as a flow proof, the three boxes across the top are LM ≅ NM (Given), ∠LMO ≅ ∠NMO (Definition of angle bisector, from MO bisects ∠LMN), and MO ≅ MO (Reflexive Property). All three arrows point down into one box: ▵MOL ≅ ▵MON (SAS).

The habit worth building here: whenever two triangles in a proof share a side or share a vertex angle, write down the reflexive statement immediately. It is almost always the piece the problem expects you to supply yourself, and once it is on the page the congruence rule usually becomes obvious.