Question

u is the midpoint of \\( \overline { s t } \\). complete the proof that \\( \angle s r u \cong \angle t r u \\).

Explanation:

Step1: Determine the reason for \( \overline{RU}\perp\overline{ST} \)

From the figure, we can see that \( \overline{RU}\perp\overline{ST} \) is given by the right - angle symbol in the diagram. So the reason is "Given".

Step2: Determine the reason for \( \angle RUS\cong\angle RUT \)

Since \( \overline{RU}\perp\overline{ST} \), by the definition of perpendicular lines (\( \perp \)), \( \angle RUS = 90^{\circ}\) and \( \angle RUT=90^{\circ}\). So \( \angle RUS\cong\angle RUT \) (All right angles are congruent).

Step3: Determine the reason for \( \overline{SU}\cong\overline{TU} \)

Because \( U \) is the midpoint of \( \overline{ST} \), by the definition of a midpoint, if \( U \) is the midpoint of a segment \( \overline{ST} \), then \( SU = TU \), so \( \overline{SU}\cong\overline{TU} \).

Step4: Determine the reason for \( \overline{RU}\cong\overline{RU} \)

By the reflexive property of congruence, any segment is congruent to itself. So \( \overline{RU}\cong\overline{RU} \) (Reflexive property of \( \cong \)).

Answer:

1. Given; 3. All right angles are congruent; 4. Definition of mid - point; 5. Reflexive property of \( \cong \)