Question

Write proofs in two-column form.

Given: Plane bisects $\overline{AB}$;$\overline{PO}\perp \overline{AB}$

Prove:$△POA\cong △POB$

Short answer

The two-column proof is:

Statements	Reasons
bisects$\overline{AB}$	Given
$\overline{AO}\cong \overline{BO}$	Definition of midpoint
$\overline{PO}\perp \overline{AB}$	Given
$m\angle POA=m\angle POB=90°$	Definition of perpendicular lines.
$\overline{PO}\cong \overline{PO}$	Reflexive property
$△POA\cong △POB$	SAS postulate

Step-by-step solution

Step 1. Observe the figure.

The figure is:

Step 2. Description of step.

Given thatplane$M$bisects $\overline{AB}$and $\overline{PO}\perp \overline{AB}$.

Therefore, by using the definition of midpoint, it can be said that $AO=OB$.

That implies, $AO\cong OB$.

As, $\overline{PO}\perp \overline{AB}$, therefore by using the definition of perpendicular lines it can be noticed that $m\angle POA=m\angle POB=90°$.

In the triangles $△POA$and $△POB$, it can be noticed that $PO=PO$by using the reflexive property.

That implies, $PO\cong PO$.

Therefore, it can be seen that $AO=OB$,$m\angle POA=m\angle POB=90°$and $PO=PO$.

Therefore, the triangles $△POA$ and $△POB$ are the congruent triangles by using the SAS postulate.

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

The proof in two-column form is:

Statements	Reasons
$M$bisects$\overline{AB}$	Given
$\overline{AO}\cong \overline{BO}$	Definition of midpoint
$\overline{PO}\perp \overline{AB}$	Given
$m\angle POA=m\angle POB=90°$	Definition of perpendicular lines.
$\overline{PO}\cong \overline{PO}$	Reflexive property
$△POA\cong △POB$	SAS postulate