Question

u is the midpoint of \\( \overline{qs} \\) and \\( \overline{rt} \\). complete the proof that \\( \triangle qtu \cong \triangle sru \\).

statement	reason
2	u is the midpoint of \\( \overline{rt} \\)	given
3	\\( \overline{qu} \cong \overline{su} \\)	definition of midpoint
4	\\( \overline{ru} \cong \overline{tu} \\)	definition of midpoint
5	\\( \angle qut \cong \angle rus \\)	vertical angle theorem
6	\\( \triangle qtu \cong \triangle sru \\)

Explanation:

Step1: Recall the Side - Angle - Side (SAS) Congruence Theorem

The SAS congruence theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify the corresponding parts in \(\triangle QTU\) and \(\triangle SRU\)

• We know that \(\overline{QU}\cong\overline{SU}\) (from step 3, definition of mid - point)
• \(\overline{TU}\cong\overline{RU}\) (from step 4, definition of mid - point)
• \(\angle QUT\cong\angle RUS\) (from step 5, vertical angle theorem)

In \(\triangle QTU\) and \(\triangle SRU\), \(\overline{QU}\) and \(\overline{SU}\), \(\overline{TU}\) and \(\overline{RU}\) are the pairs of congruent sides, and \(\angle QUT\) and \(\angle RUS\) is the pair of congruent included angles.

Answer:

By the Side - Angle - Side (SAS) Congruence Theorem.