Question

proof: given v is the midpoint of rt. given δrsv ≅ δtsv reflexive prop. sss def. of midpoint \\(\overline{rv} \cong \overline{tv}\\) \\(\overline{rv} \cong \overline{rv}\\) \\(\overline{rs} \cong \overline{ts}\\)

Explanation:

Step1: Identify Given Information

We know \( V \) is the midpoint of \( RT \), so by the definition of a midpoint, \( RV = TV \) (i.e., \( \overline{RV} \cong \overline{TV} \)). Also, from the diagram (and likely given, though not fully stated here), \( \overline{RS} \cong \overline{TS} \) (maybe given or from markings), and \( \overline{SV} \cong \overline{SV} \) by the Reflexive Property.

Step2: Apply SSS Congruence

To prove \( \triangle RSV \cong \triangle TSV \), we use the SSS (Side - Side - Side) Congruence Criterion. We have:

• \( \overline{RV} \cong \overline{TV} \) (from midpoint definition)
• \( \overline{RS} \cong \overline{TS} \) (given or from diagram markings)
• \( \overline{SV} \cong \overline{SV} \) (Reflexive Property)

So, by SSS, \( \triangle RSV \cong \triangle TSV \).

Step3: Fill in the Proof Flowchart

• The first "Given" box (top left blue) should be something like \( \overline{RS} \cong \overline{TS} \) (assuming it's given).
• The box below "V is the midpoint of RT" should be \( \overline{RV} \cong \overline{TV} \) (by definition of midpoint).
• The box for the Reflexive Property is \( \overline{SV} \cong \overline{SV} \).
• The congruence \( \triangle RSV \cong \triangle TSV \) is by SSS.

Answer:

To complete the proof:

• Top left blue "Given" box: \( \boldsymbol{\overline{RS} \cong \overline{TS}} \)
• Box below "V is the midpoint of RT": \( \boldsymbol{\overline{RV} \cong \overline{TV}} \) (by Def. of midpoint)
• Reflexive Property box: \( \boldsymbol{\overline{SV} \cong \overline{SV}} \)
• Congruence reason: \( \boldsymbol{SSS} \) (for \( \triangle RSV \cong \triangle TSV \))