Question

u is the midpoint of \\(\overline{rt}\\) and \\(\overline{qs}\\). complete the proof that \\(\triangle qtu \cong \triangle sru\\).
statement:
1 u is the midpoint of \\(\overline{rt}\\) - given
2 u is the midpoint of \\(\overline{qs}\\) - given
3 \\(\overline{ru} = \overline{tu}\\) - definition of midpoint
4 \\(\overline{qu} = \overline{su}\\) - definition of midpoint
5 \\(\angle qut = \angle rus\\) - vertical angle theorem
6 blank for statement
there is a diagram of two triangles \\(\triangle rus\\) and \\(\triangle qut\\) with vertices r, s, u and q, t, u intersecting at u

Explanation:

Step1: Recall Triangle Congruence

We need to prove \(\triangle QTU \cong \triangle SRU\). We have sides from midpoints and vertical angles.

Step2: Identify Congruence Criterion

We have \(RU = TU\) (midpoint of \(RT\)), \(QU = SU\) (midpoint of \(QS\)), and \(\angle QUT=\angle RUS\) (vertical angles). By SAS (Side - Angle - Side) congruence criterion, \(\triangle QTU \cong \triangle SRU\).

Step3: Complete the Proof Table

The last row (statement 6) should be \(\triangle QTU \cong \triangle SRU\) and the reason is "SAS Congruence Postulate" (since we have two sides and the included angle congruent).

Answer:

Statement 6: \(\triangle QTU \cong \triangle SRU\); Reason: SAS Congruence Postulate