Exercise 2 Page 278

We want to write a flow proof of the conjecture that triangles $JML$ and $MJK$ are congruent. Before we do that, let's recall what we know about flow proofs.

A $\underline{\text{flow proof}}$ uses statements written in boxes and arrows to show the logical progression of an argument. The reason justifying each statement is written below the box.

We begin by stating what we are given and what needs be the outcome of the proof. Let's now take a look at the diagram. Notice that $\overline{JM}$ is a transversal to the parallel segments $\overline{JK}$ and $\overline{LM}.$ It is also a transversal to the parallel segments $\overline{JL}$ and $\overline{KM}.$ By applying the Alternate Interior Angles Theorem we get that the two following pairs of angles are congruent. Also notice that $\overline{JM}$ is a common side for both triangles. By the Reflexive Property of Congruent Segments it is congruent to itself. Let's summarize our findings. Consequently, by applying the Angle-Side-Angle (ASA) Congruence Postulate we can conclude that $△JML$ and $△MJK$ are congruent. Let's summarize this proof using a flow chart.

Flow Proof