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$Consider the figure. Which reason can be used to justify Step 5 ? Given: bar(QR)~= bar(QT); S is the midpoint of bar(RT). Prove: /_\QRS~=/_\QTS Statements Reasons 1. bar(QR)~= bar(QT) 1. Given 2. S is the midpoint of bar(RT). 2. Given 3. bar(TS)~= bar(SR) 3. ? 4. bar(QS)~= bar(SQ) 4. ? 5. /_\QRS~=/_\QTS 5. ? A) Midpoint Theorem B) AAS C) SSS D) Reflexive Property$

Consider the figure. Which reason can be used to justify Step $5$ ? $\newline$ Given: $\overline{Q R} \cong \overline{Q T}$ ; $S$ is the midpoint of $\overline{R T}$ . $\newline$ Prove: $\triangle Q R S \cong \triangle Q T S$ $\newline$ \begin{tabular}{|l|l|} $\newline$ \hline \multicolumn{ $1$ }{|c|}{ Statements } & \multicolumn{ $1$ }{c|}{ Reasons } \\ $\newline$ \hline $1$ . $\overline{Q R} \cong \overline{Q T}$ & $1$ . Given \\ $\newline$ \hline $2$ . $S$ is the midpoint of $\overline{R T}$ . & $2$ . Given \\ $\newline$ \hline $3$ . $\overline{T S} \cong \overline{S R}$ & $3$ . ? \\ $\newline$ \hline $4$ . $\overline{Q S} \cong \overline{S Q}$ & $4$ . ? \\ $\newline$ \hline $5$ . $\triangle Q R S \cong \triangle Q T S$ & $5$ . ? \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ A) Midpoint Theorem $\newline$ B) AAS $\newline$ C) SSS $\newline$ D) Reflexive Property

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Full solution

Q. Consider the figure. Which reason can be used to justify Step

5

\newline

Given:

\overline{Q R} \cong \overline{Q T}

;

S

is the midpoint of

\overline{R T}

\newline

Prove:

\triangle Q R S \cong \triangle Q T S

\newline

\begin{tabular}{|l|l|}

\newline

\hline \multicolumn{

1

}{|c|}{ Statements } & \multicolumn{

1

}{c|}{ Reasons } \\

\newline

\hline

1

\overline{Q R} \cong \overline{Q T}

1

. Given \\

\newline

\hline

2

S

is the midpoint of

\overline{R T}

. &

2

. Given \\

\newline

\hline

3

\overline{T S} \cong \overline{S R}

3

. ? \\

\newline

\hline

4

\overline{Q S} \cong \overline{S Q}

4

. ? \\

\newline

\hline

5

\triangle Q R S \cong \triangle Q T S

5

. ? \\

\newline

\hline

\newline

\end{tabular}

\newline

A) Midpoint Theorem

\newline

B) AAS

\newline

C) SSS

\newline

D) Reflexive Property

Given Information and Proof: Look at the given information and what needs to be proved. $\newline$ Given: $\overline{QR}$ $\cong$ $\overline{QT}$ and $S$ is the midpoint of $\overline{RT}$ . $\newline$ Prove: $\angle QRS$ $\cong$ $\angle QTS$
Reason for Statement $3$ : Identify the reason for statement $3$ . $\newline$ Since $S$ is the midpoint of $\overline{RT}$ , by definition of midpoint, $\overline{TS} \cong \overline{SR}$ . $\newline$ Reason for statement $3$ : Midpoint Theorem.
Reason for Statement $4$ : Identify the reason for statement $4$ . $\overline{QS}$ is the same segment as $\overline{SQ}$ , so they are congruent by definition. Reason for statement $4$ : Reflexive Property.
Reason for Statement $5$ : Determine the reason for statement $5$ . $\newline$ We have two pairs of congruent sides and a pair of congruent angles not included between them. $\newline$ Reason for statement $5$ : This should be AAS (Angle-Angle-Side) congruence criterion, but let's check the options.
Checking Options: Check the options against the information we have. $\newline$ A) Midpoint Theorem - Already used for statement $3$ . $\newline$ B) AAS - Seems correct as we have two angles and a non-included side congruent. $\newline$ C) SSS - We don't have three sides congruent. $\newline$ D) Reflexive Property - Already used for statement $4$ . $\newline$ The correct answer is B) AAS.