Question

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complete the proof that ( overline{su} cong overline{tu} ).

Explanation:

Step1: Given

From the problem statement, we know that \(\angle SRU\cong\angle TRU\) is given.

Step2: Given

From the problem statement, we know that \(\overline{RU}\perp\overline{ST}\) is given.

Step3: Definition of perpendicular lines

Since \(\overline{RU}\perp\overline{ST}\), by the definition of perpendicular lines, \(\angle RUS\) and \(\angle RUT\) are right angles. And all right angles are congruent, so \(\angle RUS\cong\angle RUT\).

Step4: Reflexive property of congruence

For any segment \(\overline{AB}\), \(\overline{AB}\cong\overline{AB}\). So for \(\overline{RU}\), \(\overline{RU}\cong\overline{RU}\) by the reflexive property of congruence.

Step5: ASA (Angle - Side - Angle) congruence criterion

We have \(\angle SRU\cong\angle TRU\) (Angle), \(\overline{RU}\cong\overline{RU}\) (Side), \(\angle RUS\cong\angle RUT\) (Angle). So, by the ASA congruence criterion, \(\triangle RSU\cong\triangle RTU\).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle RSU\cong\triangle RTU\), their corresponding parts \(\overline{SU}\) and \(\overline{TU}\) are congruent. So \(\overline{SU}\cong\overline{TU}\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (all right angles are congruent)
4. Reflexive property of congruence
5. ASA (Angle - Side - Angle)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)