Question

the proof that $\triangle qpt \cong \triangle qrt$ is shown.
given: $sp \cong sr$
line segment $qt$
prove: $\triangle qpt \cong \triangle qrt$

1. $\overline{st} \perp \overline{pr}$
2. converse of the perpendicular bisector theorem
3. $\overline{pt} \cong \overline{rt}$
4. ?
5. $\overline{qt} \perp \overline{pr}$
6. $\overline{st}$ and $\overline{qt}$ name the same line.
7. $\overline{qp} \cong \overline{qr}$
8. perpendicular bisector theorem
9. $\triangle qpt \cong \triangle qrt$
10. hl theorem

(substitution property, definition of perpendicular bisector, definition of congruence, reflexive property)

Explanation:

Step1: Analyze the Given Information

We know that \( SP \cong SR \) (given) and \( ST \perp PR \) (from step 2). A perpendicular bisector of a segment is a line that is perpendicular to the segment and bisects it (divides it into two congruent parts). Since \( ST \perp PR \) and \( SP \cong SR \), \( T \) should be the midpoint of \( PR \) (by the definition of perpendicular bisector, which states that if a line is perpendicular to a segment and passes through a point equidistant from the endpoints of the segment, it bisects the segment). So, \( PT \cong RT \) because \( T \) bisects \( PR \).

Step2: Identify the Justification

The reason for \( PT \cong RT \) is the definition of a perpendicular bisector. A perpendicular bisector of a segment bisects the segment, meaning it divides the segment into two congruent segments. Since \( ST \) is the perpendicular bisector of \( PR \) (as \( ST \perp PR \) and \( SP = SR \)), it follows that \( PT = RT \).

Answer:

definition of perpendicular bisector