Question

fill in the blank 1 point
complete the proof.
copy the diagram to your show your work page to mark congruent segments or angles.
given: ( s ) is the midpoint of ( overline{qu}, overline{qr} cong overline{st}, overline{rs} cong overline{tu} )
prove: ( \triangle qrs cong \triangle stu )

Explanation:

Step1: Recall the Side - Side - Side (SSS) Congruence Criterion

The SSS congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
We know that \( \overline{QR}\cong\overline{ST}\), \( \overline{RS}\cong\overline{TU}\), and from the fact that \(S\) is the mid - point of \( \overline{QU}\), we have \( \overline{QS}\cong\overline{SU}\) (by the definition of a mid - point, which means \(QS = SU\) and thus the segments are congruent).

Step2: Apply the SSS Congruence Criterion

In \( \triangle QRS\) and \( \triangle STU\), we have:

• \( \overline{QR}\cong\overline{ST}\) (given)
• \( \overline{RS}\cong\overline{TU}\) (given)
• \( \overline{QS}\cong\overline{SU}\) (from the mid - point definition)

By the SSS (\(SSS\)) congruence criterion, \( \triangle QRS\cong\triangle STU\)

Answer:

1. \( \overline{QS}\cong\overline{SU}\)
2. \(SSS\) (Side - Side - Side) Congruence Criterion