Question

given: $overline{wt}congoverline{tu}$; $overline{ru}$ and $overline{sw}$ are medians of $\triangle wtu$. prove: $overline{ru}congoverline{sw}$ 8. $\triangle rwucong\triangle suw$ 9. $overline{ru}congoverline{sw}$

Explanation:

Step1: Recall median property

Since $\overline{RU}$ and $\overline{SW}$ are medians of $\triangle WTU$, $R$ is the mid - point of $\overline{WT}$ and $S$ is the mid - point of $\overline{TU}$. Given $\overline{WT}\cong\overline{TU}$, then $\overline{WR}\cong\overline{TS}$ and $\overline{RT}\cong\overline{SU}$ (by the definition of mid - point and congruent segments). Also, $\overline{WU}\cong\overline{UW}$ (reflexive property of congruence).

Step2: Prove triangle congruence

In $\triangle RWU$ and $\triangle SUW$, we have $\overline{WR}\cong\overline{TS}$, $\overline{WU}\cong\overline{UW}$, and $\overline{RT}\cong\overline{SU}$. By the Side - Side - Side (SSS) congruence criterion, $\triangle RWU\cong\triangle SUW$.

Step3: Use congruent triangle property

If $\triangle RWU\cong\triangle SUW$, then corresponding parts of congruent triangles are congruent. So, $\overline{RU}\cong\overline{SW}$ (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Statements: Since $\overline{RU}$ and $\overline{SW}$ are medians, $\overline{WR}\cong\overline{RT}$ and $\overline{TS}\cong\overline{SU}$; $\overline{WT}\cong\overline{TU}$ (Given), so $\overline{WR}\cong\overline{TS}$ and $\overline{RT}\cong\overline{SU}$; $\overline{WU}\cong\overline{UW}$ (Reflexive property). Reasons: Definition of median, given, reflexive property of congruence.
2. Statements: $\triangle RWU\cong\triangle SUW$. Reasons: SSS (Side - Side - Side) congruence criterion.
3. Statements: $\overline{RU}\cong\overline{SW}$. Reasons: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).