Question

r is the midpoint of qs. complete the proof that ∠u ≅ ∠t.
1 r is the midpoint of overline{qs} given
2 overline{rt} ≅ overline{ru} given
3 overline{su} ≅ overline{qt} given
4 overline{qr} ≅ overline{rs} definition of midpoint
5

Explanation:

Step1: Use SSS (Side - Side - Side) Congruence Criterion

We know that in \(\triangle QRT\) and \(\triangle SRU\):

• \(\overline{QR}\cong\overline{RS}\) (from the definition of mid - point, as \(R\) is the mid - point of \(QS\))
• \(\overline{RT}\cong\overline{RU}\) (given)
• \(\overline{QT}\cong\overline{SU}\) (given)

So, \(\triangle QRT\cong\triangle SRU\) by the SSS (Side - Side - Side) congruence criterion.

Step2: Use Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Since \(\triangle QRT\cong\triangle SRU\), then \(\angle U\cong\angle T\) (by CPCTC, which states that if two triangles are congruent, then their corresponding parts are congruent)

Answer:

1. \(\triangle QRT\cong\triangle SRU\), SSS Congruence Criterion; \(\angle U\cong\angle T\), CPCTC.