Question

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

1 u is the midpoint of \overline{qs}
2 \overline{qt} \perp \overline{qs}
3 \overline{rs} \perp \overline{qs}
4 \angle q \cong \angle s
5 \overline{qu} \cong \overline{su}
6 \angle qut \cong \angle rus
7 \triangle qtu \cong \triangle sru

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Definition of perpendicular

Since \(\overline{QT}\perp\overline{QS}\), by the definition of perpendicular, \(\angle Q = 90^{\circ}\).

Step3: Definition of perpendicular

Since \(\overline{RS}\perp\overline{QS}\), by the definition of perpendicular, \(\angle S=90^{\circ}\).

Step4: All right angles are congruent

Since \(\angle Q = 90^{\circ}\) and \(\angle S = 90^{\circ}\), \(\angle Q\cong\angle S\) (all right angles are congruent).

Step5: Definition of mid - point

Since \(U\) is the mid - point of \(\overline{QS}\), by the definition of mid - point, \(QU = SU\), so \(\overline{QU}\cong\overline{SU}\).

Step6: Vertical angles are congruent

\(\angle QUT\) and \(\angle RUS\) are vertical angles. By the vertical angles theorem, \(\angle QUT\cong\angle RUS\).

Step7: ASA (Angle - Side - Angle) congruence criterion

In \(\triangle QTU\) and \(\triangle SRU\), we have \(\angle Q\cong\angle S\), \(\overline{QU}\cong\overline{SU}\), \(\angle QUT\cong\angle RUS\). By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle QTU\cong\triangle SRU\).

Answer:

1. Given
2. Definition of perpendicular
3. Definition of perpendicular
4. All right angles are congruent
5. Definition of mid - point
6. Vertical angles are congruent
7. ASA (Angle - Side - Angle) congruence criterion