Question

given: s is the mid - point of qu, qr = st, rs = tu. prove: △qrs≅△stu. statements: 1. s is the midpoint of qu. 2. qr = st. 3. rs = tu. 4. qs = su. 5. △qrs≅△stu. reasons: 1. given. 2. given. 3. given. 4. 5.

Explanation:

Step1: Define mid - point property

Since \(S\) is the mid - point of \(\overline{QU}\), by the definition of a mid - point, it divides the line segment \(\overline{QU}\) into two congruent segments, so \(\overline{QS}\cong\overline{SU}\).

Step2: Apply congruence criterion

We have \(\overline{QR}\cong\overline{ST}\), \(\overline{RS}\cong\overline{TU}\), and \(\overline{QS}\cong\overline{SU}\). By the Side - Side - Side (SSS) congruence criterion for triangles, which states that if the three sides of one triangle are congruent to the three corresponding sides of another triangle, then the two triangles are congruent. So, \(\triangle QRS\cong\triangle STU\).

Answer:

1. Definition of mid - point; 5. Side - Side - Side (SSS) congruence criterion