T is the midpoint of overline(R S) and overline(R S) perp overline(Q T) . Complete the proof that overline(Q R) cong overline(Q S) . & Statement & Reason 1 & T is the midpoint of overline(R S) & Given 2 & overline(R S) perp overline(Q T) & Given 3 & angle Q T R cong angle Q T S & All right angles are congruent 4 & overline(R T) cong overline(S T) & Definition of midpoint 5 & overline(Q T) cong overline(Q T) & Reflexive Property of Congruence 6 & triangle Q R T cong triangle Q S T & 7 & overline(Q R) cong overline(Q S) &

To complete the proof that \(\overline{Q R} \cong \overline{Q S}\), we need to fill in the missing reasons for statements 6 and 7 in the proof table. Here are the steps to complete the proof:Step 1: Given that \(T\) is the midpoint of \(\overline{R S}\), we can say that \(\overline{R T} \cong \overline{S T}\) (Statement 4) because the definition of a midpoint is that it divides a segment into two congruent segments.Step 2: Given that \(\overline{R S} \perp \overline{Q T}\), we know that \(\angle Q T R\) and \(\angle Q T S\) are right angles (Statement 3) because perpendicular lines form right angles.Step 3: We can also say that \(\overline{Q T} \cong \overline{Q T}\) (Statement 5) by the Reflexive Property of Congruence, which states that any geometric figure is congruent to itself.Step 4: Now, we have two triangles, \(\Delta Q R T\) and \(\Delta Q S T\), with the following congruences:- \(\angle Q T R \cong \angle Q T S\) (from step 2)- \(\overline{R T} \cong \overline{S T}\) (from step 1)- \(\overline{Q T} \cong \overline{Q T}\) (from step 3)Step 5: With these congruences, we can use the Hypotenuse-Leg (HL) Congruence Theorem, which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Therefore, \(\Delta Q R T \cong \Delta Q S T\) (Statement 6) by the HL Congruence Theorem.Step 6: By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, if two triangles are congruent, then all of their corresponding parts are congruent. Since \(\Delta Q R T \cong \Delta Q S T\), we can say that \(\overline{Q R} \cong \overline{Q S}\) (Statement 7) by CPCTC.Final Answer:\begin{array}{|l|l|l|}\hline\ &\ Statement\ &\ Reason\ \\\hline\ 1\ &\ T\ is\ the\ midpoint\ of\ \overline{R\ S}\ &\ Given\ \\2\ &\ \overline{R\ S}\ \perp\ \overline{Q\ T}\ &\ Given\ \\3\ &\ \angle\ Q\ T\ R\ \cong\ \angle\ Q\ T\ S\ &\ All\ right\ angles\ are\ congruent\ \\4\ &\ \overline{R\ T}\ \cong\ \overline{S\ T}\ &\ Definition\ of\ midpoint\ \\5\ &\ \overline{Q\ T}\ \cong\ \overline{Q\ T}\ &\ Reflexive\ Property\ of\ Congruence\ \\6\ &\ \Delta\ Q\ R\ T\ \cong\ \Delta\ Q\ S\ T\ &\ HL\ Congruence\ Theorem\ \\7\ &\ \overline{Q\ R}\ \cong\ \overline{Q\ S}\ &\ CPCTC\ \\\hline\end{array}