Question

Write proofs in two-column form.

Given: $\overline{TM}\cong \overline{PR}$;$\overline{TM}\parallel \overline{RP}$

Prove:$△TEM\cong △PER$

Short answer

The two-column proof is:

Statements	Reasons
$\overline{TM}\parallel \overline{RP}$	Given
$\angle ETM=\angle EPR,\angle TME=\angle PRE$	Alternate interior angles
$\overline{TM}\cong \overline{PR}$	Given
$△TEM\cong △PER$	ASA Postulate

Step-by-step solution

Step 1. Observe the figure.

The figure is:

Step 2. Description of step.

Given that$\overline{TM}\cong \overline{PR}$and $\overline{TM}\parallel \overline{RP}$.

As, $\overline{TM}\parallel \overline{RP}$and from the given diagram it can be noticed that the angles $\angle ETM$ and $\angle EPR$ are the alternate interior angles.

Therefore, $\angle ETM=\angle EPR$.

That implies, $\angle ETM\cong \angle EPR$.

As, $\overline{TM}\parallel \overline{RP}$and from the given diagram it can be noticed that the angles $\angle TME$ and $\angle ERP$ are the alternate interior angles.

Therefore, $\angle TME=\angle PRE$.

That implies, $\angle TME\cong \angle PRE$.

Therefore, it can be seen that $\angle ETM=\angle EPR$, $\overline{TM}=\overline{PR}$and $\angle TME=\angle PRE$.

Therefore, the triangles $△TEM$ and $△PER$ are the congruent triangles by using the ASA postulate.

Step 3. Write the proof in two-column form.

The proof in two-column form is:

Statements	Reasons
$\overline{TM}\parallel \overline{RP}$	Given
$\angle ETM=\angle EPR$,$\angle TME=\angle PRE$	Alternate interior angles
$\overline{TM}\cong \overline{PR}$	Given
$△TEM\cong △PER$	ASA Postulate