Question

t is the midpoint of \overline{su}, \triangle qrt is equilateral, and \overline{rs} \cong \overline{qu}. complete the proof that \angle srt \cong \angle tqu.

1 t is the midpoint of \overline{su}
2 \triangle qrt is equilateral
3 \overline{rs} \cong \overline{qu}
4 \overline{tu} \cong \overline{st}
5 \overline{qt} \cong \overline{rt}
6

Explanation:

Step1: Definition of midpoint

Since \(T\) is the midpoint of \(\overline{SU}\), by the definition of a midpoint, \( \overline{TU}\cong\overline{ST}\).

Step2: Properties of equilateral triangle

Since \(\triangle QRT\) is equilateral, all its sides are equal. So, \( \overline{QT}\cong\overline{RT}\).

Step3: SSS (Side - Side - Side) Congruence Criterion

We have \(\overline{RS}\cong\overline{QU}\) (given), \(\overline{TU}\cong\overline{ST}\) (from step 1), and \(\overline{QT}\cong\overline{RT}\) (from step 2).
By the SSS congruence criterion, \(\triangle SRT\cong\triangle TQU\).

Step4: Corresponding parts of congruent triangles

If \(\triangle SRT\cong\triangle TQU\), then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle SRT\cong\angle TQU\).

Answer:

\(\angle SRT\cong\angle TQU\) because \(\triangle SRT\cong\triangle TQU\) (by SSS) and CPCTC.