Question

the proof that △qpt≅△qrt is shown. given: $overline{sp}congoverline{sr}$, line segment $overline{qt}$. prove: △qpt≅△qrt. what is the missing reason in the proof? statements: 1. $overline{sp}congoverline{sr}$, reason: 1. given; 2. $overline{st}perpoverline{pr}$, reason: 2. converse of the perpendicular bisector theorem; 3. $overline{pt}congoverline{rt}$, reason: 3. ?; 4. $overline{qt}perpoverline{pr}$, reason: 4. $overline{st}$ and $overline{qt}$ name the same line.; 5. $overline{qp}congoverline{qr}$, reason: 5. perpendicular bisector theorem; 6. △qpt≅△qrt, reason: 6. hl theorem.

Explanation:

Step1: Recall perpendicular bisector property

If a point lies on the perpendicular bisector of a line - segment, then it is equidistant from the endpoints of the line - segment.
Since $\overline{ST}\perp\overline{PR}$ and it is related to the perpendicular bisector concept (from step 2), and we want to show $\overline{PT}\cong\overline{RT}$. The point $T$ lies on the perpendicular bisector of $\overline{PR}$ (implied by $\overline{ST}\perp\overline{PR}$ and relevant geometric relationships).

Step2: Identify the reason

The reason for $\overline{PT}\cong\overline{RT}$ is the definition of a perpendicular bisector (a point on the perpendicular bisector of a line segment is equidistant from the segment's endpoints).

Answer:

definition of a perpendicular bisector