Questions: ST is perpendicular to RS, RU is perpendicular to TU, and angle TRU is congruent to angle RTS. Complete the proof that triangle RTU is congruent to triangle TRS. 1. ST is perpendicular to RS - Given 2. RU is perpendicular to TU - Given 3. angle TRU is congruent to angle RTS - Given 4. angle S is congruent to angle U 5. RT is congruent to RT - Reflexive Property of Congruence 6. triangle RTU is congruent to triangle TRS

ST is perpendicular to RS, RU is perpendicular to TU, and angle TRU is congruent to angle RTS. Complete the proof that triangle RTU is congruent to triangle TRS. 1. ST is perpendicular to RS - Given 2. RU is perpendicular to TU - Given 3. angle TRU is congruent to angle RTS - Given 4. angle S is congruent to angle U 5. RT is congruent to RT - Reflexive Property of Congruence 6. triangle RTU is congruent to triangle TRS
Transcript text: $\overline{S T} \perp \overline{R S}, \overline{R U} \perp \overline{T U}$, and $\angle T R U \cong \angle R T S$. Complete the proof that $\triangle R T U \cong \triangle T R S$. \begin{tabular}{|l|l|l} \hline & Statement & Reason \\ \hline 1 & $\overline{S T} \perp \overline{R S}$ & Given \\ 2 & $\overline{R U} \perp \overline{T U}$ & Given \\ 3 & $\angle T R U \cong \angle R T S$ & Given \\ 4 & $\angle S \cong \angle U$ & \\ 5 & $\overline{R T} \cong \overline{R T}$ & Reflexive Property of Congruence \\ 6 & $\triangle R T U \cong \triangle T R S$ & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Given Information
  • \( \overline{ST} \perp \overline{RS} \)
  • \( \overline{RU} \perp \overline{TU} \)
  • \( \angle TRU \cong \angle RTS \)
Step 2: Identify Congruent Angles
  • \( \angle S \cong \angle U \)
    • Reason: Both angles are right angles (90 degrees) because \( \overline{ST} \perp \overline{RS} \) and \( \overline{RU} \perp \overline{TU} \).
Step 3: Reflexive Property
  • \( \overline{RT} \cong \overline{RT} \)
    • Reason: Reflexive Property of Congruence (a segment is congruent to itself).

Final Answer

  • \( \triangle RTU \cong \triangle TRS \)
    • Reason: By the Angle-Angle-Side (AAS) Congruence Postulate, since two angles and the non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle.
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