Question

question 7
justify the last two steps of the proof.
given: \\( \overline { r s } \cong \overline { u t } \\) and \\( \overline { r t } \cong \overline { u s } \\)
prove: \\( \triangle r s t \cong \triangle u t s \\)
proof

1. \\( \overline { r s } \cong \overline { u t } \\) 1. given
2. \\( \overline { r t } \cong \overline { u s } \\) 2. given
3. \\( s t = s t \\)
4. triangle rst = triangle uts

Explanation:

Step1: Justify \( ST = ST \)

Reflexive Property of Congruence states that any segment is congruent to itself. So, \( ST = ST \) is based on this property.

Step2: Justify \( \triangle RST\cong\triangle UTS \)

We know that \( \overline{RS}\cong\overline{UT}\), \( \overline{RT}\cong\overline{US}\), and \( \overline{ST}\cong\overline{ST}\). The SSS (Side - Side - Side) Congruence Postulate says that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, \( \triangle RST\cong\triangle UTS\) by SSS.

Answer:

1. \( ST = ST \) is justified by the Reflexive Property of Congruence.
2. \( \triangle RST\cong\triangle UTS \) is justified by the SSS (Side - Side - Side) Congruence Postulate.