Question

given: rectangle qprs,
\\( \overline { q u } \cong \overline { s u } \\)
prove: \\( \overline { p v } \cong \overline { r t } \\)

1. rectangle qprs,

\\( \overline { q u } \cong \overline { s u } \\)

1. click here to insert
2. click here to insert
3. \\( \angle p v q \cong \angle u q s \\),

\\( \angle r t s \cong \angle u s q \\)

1. given
2. base angles theorem
3. definition of a rec -

tangle/parallelogram

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Explanation:

Step1: Base Angles Theorem

Since \( \overline{QU}\cong\overline{SU}\), then \( \angle UQS\cong\angle USQ\) (Base Angles Theorem: In an isosceles triangle, the angles opposite the congruent sides are congruent).

Step2: Properties of a Rectangle

In rectangle \( QPRS\), \( \overline{PQ}\parallel\overline{RS}\) (Definition of a rectangle: A rectangle is a parallelogram, and in a parallelogram, opposite sides are parallel).

Step3: Alternate - Interior Angles

Because \( \overline{PQ}\parallel\overline{RS}\), \( \angle PVQ\cong\angle UQS\) (Alternate - interior angles: If two parallel lines are cut by a transversal, then alternate - interior angles are congruent) and \( \angle RTS\cong\angle USQ\) (Alternate - interior angles).

Step4: Transitive Property

Since \( \angle UQS\cong\angle USQ\), \( \angle PVQ\cong\angle UQS\), and \( \angle RTS\cong\angle USQ\), by the transitive property of congruence \( \angle PVQ\cong\angle RTS\).

Step5: AAS (Angle - Angle - Side) Congruence

In \( \triangle PVQ\) and \( \triangle RTS\), \( \angle PVQ\cong\angle RTS\), \( \angle QPV\cong\angle SR T\) (All angles in a rectangle are right angles, so \( \angle QPV = \angle SRT=90^{\circ}\)), and \( \overline{PQ}\cong\overline{RS}\) (Opposite sides of a rectangle are congruent). So, \( \triangle PVQ\cong\triangle RTS\) (AAS congruence: If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle PVQ\cong\triangle RTS\), then \( \overline{PV}\cong\overline{RT}\) (CPCTC: If two triangles are congruent, then their corresponding parts are congruent).

Answer:

\(\overline{PV}\cong\overline{RT}\)