Question

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ru bisects ∠srt and rt ≅ rs. complete the proof that ∠rut ≅ ∠rus.

1 ru bisects ∠srt
2 rt ≅ rs
3 ∠sru ≅ ∠tru
4 ru ≅ ru
5 △rsu ≅ △rtu
6 ∠rut ≅ ∠rus

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem.

Step3: Definition of angle bisector

Since \(\overleftrightarrow{RU}\) bisects \(\angle SRT\), by the definition of an angle bisector, \(\angle SRU\cong\angle TRU\).

Step4: Reflexive property of congruence

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

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

We have \(\overline{RT}\cong\overline{RS}\) (from step 2), \(\angle SRU\cong\angle TRU\) (from step 3), and \(\overline{RU}\cong\overline{RU}\) (from step 4). So, \(\triangle RSU\cong\triangle RTU\) by the SAS congruence criterion.

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

Since \(\triangle RSU\cong\triangle RTU\), then \(\angle RUT\cong\angle RUS\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)