$6$ ) $\ast$ $5$ points In quadrilateral $QRST$ , diagonals $QS$ and $\overline{RT}$ intersect at $M$ . Which statement would always prove quadrilateral $QRST$ is a parallelogram? $\newline$ $1$ / $\angle TQR$ and $\angle QRS$ are supplementary.

Full solution

6

)

\ast

5

points In quadrilateral

QRST

, diagonals

QS

and

\overline{RT}

intersect at

M

. Which statement would always prove quadrilateral

QRST

is a parallelogram?

\newline

1

\angle TQR

and

\angle QRS

are supplementary.

Properties of Parallelogram: Understand the properties of a parallelogram. $\newline$ A quadrilateral is a parallelogram if both pairs of opposite sides are parallel. One way to prove this is by showing that one pair of opposite angles are supplementary, which means they add up to $180$ degrees. This is because if one pair of opposite angles are supplementary, then both pairs of opposite sides are parallel by the consecutive angles test.
Analyzing the Given Statement: Analyze the given statement. $\newline$ The statement given is that angle $TQR$ and angle $QRS$ are supplementary. If these angles are supplementary, it means that angle $TQR + QRS = 180$ degrees.
Applying Consecutive Angles Test: Apply the consecutive angles test. If angle $TQR$ and angle $QRS$ are supplementary, then by the consecutive angles test, side $QR$ must be parallel to side $TS$ , and side $QS$ must be parallel to side $RT$ . This is because the consecutive angles formed by the intersection of the diagonals with the sides of the quadrilateral are supplementary, which is a property of parallelograms.
Concluding the Proof: Conclude the proof. $\newline$ Since we have shown that one pair of opposite sides $QR$ and $TS$ are parallel, and the other pair of opposite sides $QS$ and $RT$ are also parallel, quadrilateral $QRST$ must be a parallelogram by definition.