Question

1. proof: write a two - column proof.

given: rstv is a rectangle
and u is the midpoint of \\( \overline { v t } \\).
prove: \\( \triangle r u v \cong \triangle s u t \\)
statements
\\( \begin{array} { l } { \text { (1) } r s t v \text { is a rectangle } } \\ { \text { and } u \text { is the midpoint } } \\ { \text { of } \overline { v t } } end{array} \\)
(2) \\( \overline { r v } \cong \overline { s t } \\)
(3) \\( \angle r v t \cong \angle s t v \\)
(4) \\( \overline { v u } \cong \overline { u t } \\)
(5) \\( \triangle r u v \cong \triangle s u t \\)
reasons
(1) given
(2)
(3)
(4) definition of midpoint
(5) sas

Explanation:

Step1: Properties of rectangle

In a rectangle \(RSTV\), opposite sides are congruent. So, \(\overline{RV}\cong\overline{ST}\) (by the property of a rectangle: opposite sides of a rectangle are congruent).

Step2: Angles in rectangle

In a rectangle \(RSTV\), all angles are right - angles. So, \(\angle RVT\cong\angle STV\) (all angles of a rectangle are \(90^{\circ}\), and angles with equal measure are congruent).

Step3: Mid - point property

Since \(U\) is the mid - point of \(\overline{VT}\), by the definition of a mid - point, \(\overline{VU}\cong\overline{UT}\).

Step4: SAS (Side - Angle - Side) criterion

We have \(\overline{RV}\cong\overline{ST}\), \(\angle RVT\cong\angle STV\), and \(\overline{VU}\cong\overline{UT}\). By the SAS (Side - Angle - Side) congruence criterion for triangles (\(SAS\): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle RUV\cong\triangle SUT\).

Answer:

Statements	Reasons
\(\overline{RV}\cong\overline{ST}\)	Opposite sides of a rectangle are congruent
\(\angle RVT\cong\angle STV\)	All angles of a rectangle are \(90^{\circ}\) (right angles are congruent)
\(\overline{VU}\cong\overline{UT}\)	Definition of mid - point
\(\triangle RUV\cong\triangle SUT\)	\(SAS\) (Side - Angle - Side) congruence criterion