Question

explain whether there is enough information given in the figure to prove that the triangles are congruent using sss or sas. select choice it is select choice $overline{qt} cong overline{sr}$, so you select choice use sss, and select choice of the angles are known to be congruent, so you select choice use sas.

Explanation:

Step1: Analyze the figure

From the figure, we can see that \( QS \) is a common side to both triangles \( \triangle QTS \) and \( \triangle QRS \). Also, the markings on \( QT \) and \( SR \) (and \( QR \) and \( TS \) if we assume the other pair) suggest some equal sides. Given \( \overline{QT}\cong\overline{SR} \), and \( \overline{QS}\cong\overline{QS} \) (common side). Also, if we look at the other pair of sides, from the figure, \( \overline{QR}\cong\overline{TS} \) (marked with the same tick marks). So for SSS: we have three pairs of congruent sides. For SAS: we would need two sides and the included angle. But here, with \( \overline{QT}\cong\overline{SR} \), \( \overline{QS}\cong\overline{QS} \), and \( \overline{QR}\cong\overline{TS} \), SSS is applicable. Also, if we consider the angles, but since we have three sides, SSS is sufficient. So first, it is enough information. Then, \( \overline{QT}\cong\overline{SR} \), \( \overline{QS}\cong\overline{QS} \), and \( \overline{QR}\cong\overline{TS} \) (from the figure's markings), so we can use SSS. And since no angles are known to be congruent (the problem states "of the angles are known to be congruent" but in reality, with SSS we don't need angles), so the answer for the first select choice: "Yes" (it is enough), second: "Given" ( \( \overline{QT}\cong\overline{SR} \) is given), third: "can" (you can use SSS), fourth: "None" (of the angles are known to be congruent, so we use SSS not SAS), fifth: "can" (wait, no, let's re - evaluate. Wait, the problem has dropdowns: first "it is" [Yes/No], second "Select Choice" (maybe "Given" that \( \overline{QT}\cong\overline{SR} \)), third "use SSS, and" [can/cannot], fourth "of the angles are known to be congruent, so you" [can/cannot] use SAS.

Wait, let's re - structure:

1. Is there enough info? Yes, because we have \( \overline{QT}\cong\overline{SR} \), \( \overline{QS} \) is common, and \( \overline{QR}\cong\overline{TS} \) (from the figure's markings). So first choice: Yes.

1. \( \overline{QT}\cong\overline{SR} \) is given, so second choice: Given.

1. Can we use SSS? Yes, because we have three pairs of congruent sides (\( \overline{QT}\cong\overline{SR} \), \( \overline{QS}\cong\overline{QS} \), \( \overline{QR}\cong\overline{TS} \)), so third choice: can.

1. Of the angles: None are known to be congruent (since we are using SSS, we don't need angle congruence), so fourth choice: None.

1. So we use SSS, so "can" (wait, the fifth dropdown: "so you [can/cannot] use SAS". Since we can use SSS, and no angles are known, we cannot use SAS because SAS needs two sides and included angle, and we don't have the included angle. Wait, maybe the dropdowns are:

1. It is: Yes

1. (the one with \( \overline{QT}\cong\overline{SR} \)): Given

1. use SSS, and: can

1. of the angles are known to be congruent: None

1. so you: cannot (use SAS, because we use SSS)

Step2: Fill in the dropdowns

• First dropdown: Yes (it is enough information)
• Second dropdown: Given ( \( \overline{QT}\cong\overline{SR} \) is given)
• Third dropdown: can (you can use SSS)
• Fourth dropdown: None (of the angles are known to be congruent)
• Fifth dropdown: cannot (so you cannot use SAS, you use SSS)

Answer:

First Select Choice: Yes
Second Select Choice: Given
Third Select Choice: can
Fourth Select Choice: None
Fifth Select Choice: cannot

(Note: The exact wording of the dropdown options might vary, but based on the analysis, these are the logical choices. The key is that with the given side congruence and the common side and the other pair of sides from the figure, SSS is applicable, so there is enough info, we can use SSS, and no angles are needed for SSS so we don't use SAS.)