Question

the proof that δqpt ≅ δqrt is shown. given: (overline{sp} cong overline{sr}) prove: δqpt ≅ δqrt diagram of triangle qpr with points p, t, r on the base, s and q above what is the missing reason in the proof? statements reasons 1. (overline{sp} cong overline{sr}) 1. given 2. (overline{st} perp overline{pr}) 2. converse of the perpendicular bisector theorem 3. (overline{pt} cong overline{rt}) 3. ? 4. (overline{qt} perp overline{pr}) 4. (overline{st}) and (overline{qt}) name the same line. 5. (overline{qp} cong overline{qr}) 5. perpendicular bisector theorem 6. δqpt ≅ δqrt 6. hl theorem options: definition of perpendicular bisector, definition of congruence, reflexive property, substitution property

Explanation:

Step1: Recall Perpendicular Bisector Definition

A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two congruent segments.

Step2: Analyze Given Information

We know \( \overline{ST} \perp \overline{PR} \) (from step 2) and \( \overline{SP} \cong \overline{SR} \) (given). By the definition of a perpendicular bisector, a line that is perpendicular to a segment and passes through the midpoint (since \( SP = SR \), \( S \) is on the perpendicular bisector) will bisect the segment, meaning \( \overline{PT} \cong \overline{RT} \).

Step3: Evaluate Other Options

• "Definition of congruence" refers to equal measures, not segment bisecting.
• "Reflexive property" is for a segment being congruent to itself, not applicable here.
• "Substitution property" involves replacing a value, not relevant to segment bisecting.

Answer:

definition of perpendicular bisector