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GIVEN: $\newline$ $QTVW$ is a rectangle, $\newline$ $\overline{QR} \cong \overline{TS}$ $\newline$ PROVE: $\newline$ $\triangle SWQ \cong \triangle RVT$

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Evaluate a nonlinear function

Full solution

Q. GIVEN:

\newline

QTVW

is a rectangle,

\newline

\overline{QR} \cong \overline{TS}

\newline

PROVE:

\newline

\triangle SWQ \cong \triangle RVT

Identify Properties: Identify the properties of a rectangle that are relevant to the problem. In a rectangle, opposite sides are equal and all angles are right angles ( $90^\circ$ ). Since $QTVW$ is a rectangle, we know that angle $QWT$ and angle $TVW$ are right angles.
Use Congruence Fact: Use the fact that $\overline{QR}$ is congruent to $\overline{TS}$ to determine the relationship between the triangles formed. Since $QR$ and $TS$ are congruent, and $QW$ and $TV$ are both sides of the rectangle (hence, they are equal), and we know that angle $QWT$ and angle $TVW$ are right angles, we can say that triangle $QWR$ and triangle $TSV$ are right triangles and they are congruent by the Hypotenuse-Leg (HL) theorem.
Determine Angle Relationship: Determine the relationship between the angles we need to prove congruent. Since triangle $QWR$ is congruent to triangle $TSV$ , all corresponding parts of congruent triangles are congruent (CPCTC). Therefore, angle $WQR$ is congruent to angle $VTS$ .
Relate Congruent Angles: Relate the congruent angles to the ones we need to prove congruent. Angle $SWQ$ is an extension of angle $WQR$ , and angle $RVT$ is an extension of angle $VTS$ . Since angle $WQR$ is congruent to angle $VTS$ , and straight lines create supplementary angles ( $180$ degrees), angle $SWQ$ is congruent to angle $RVT$ because they are both supplements to congruent angles ( $WQR$ and $VTS$ ).

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