Question

shawna was asked to prove that the two triangles (left) are congruent, and was given the following table of statements. provide reasons that justify each statement as true, thus proving the two triangles congruent. given: s is the midpoint of $overline{bt}$; $overline{bo} parallel overline{at}$ prove: $\triangle bos cong \triangle tas$

Explanation:

1. $\angle BSO \cong \angle TSA$: These are opposite angles formed by intersecting lines, so they are vertical angles, which are always congruent.
2. $S$ is the midpoint of segment $BT$: This is a given piece of information provided to start the proof.
3. segment $SB \cong$ segment $ST$: By definition, a midpoint divides a segment into two congruent parts.
4. segment $BO$ is parallel to segment $AT$: This is a given piece of information provided to start the proof.
5. $\angle B \cong \angle T$: When a transversal intersects two parallel lines, alternate interior angles are congruent. Here, transversal $BT$ cuts parallel lines $BO$ and $AT$, making these alternate interior angles.
6. $\triangle BOS \cong \triangle TAS$: We have two pairs of congruent angles ($\angle BSO \cong \angle TSA$, $\angle B \cong \angle T$) and a pair of congruent included sides ($SB \cong ST$), so the Angle-Side-Angle (ASA) congruence criterion applies.

Answer:

Statement	Reason
$S$ is the midpoint of segment $BT$	Given
segment $SB \cong$ segment $ST$	Definition of midpoint
segment $BO$ is parallel to segment $AT$	Given
$\angle B \cong \angle T$	Alternate interior angles (parallel lines)
$\triangle BOS \cong \triangle TAS$	ASA Congruence Criterion