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State the slopes of the parellel lines in the parellegram.

Quadrilateral QRST has vertices at Q (-1, 5), R (2, 8), S (4, 4), and T (1, 1). The measure of Angle Q is 108° and the measure of angle R is 108 - 2x.

1 answer

To find the slopes of the parallel lines in the parallelogram formed by the quadrilateral QRST, we first need to determine the slopes of the sides QR, RS, ST, and TQ.

Step 1: Calculate the slopes of each side

  1. Slope of QR: \[ \text{slope}_{QR} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 5}{2 - (-1)} = \frac{3}{3} = 1 \]

  2. Slope of RS: \[ \text{slope}_{RS} = \frac{4 - 8}{4 - 2} = \frac{-4}{2} = -2 \]

  3. Slope of ST: \[ \text{slope}_{ST} = \frac{1 - 4}{1 - 4} = \frac{-3}{-3} = 1 \]

  4. Slope of TQ: \[ \text{slope}_{TQ} = \frac{5 - 1}{-1 - 1} = \frac{4}{-2} = -2 \]

Step 2: Identify the pairs of parallel sides

In a parallelogram, opposite sides are parallel, and thus have the same slope:

  • Sides QR and ST have the same slope of 1.
  • Sides RS and TQ have the same slope of -2.

Conclusion

The slopes of the parallel lines in the parallelogram are:

  • Slope of QR and ST: 1
  • Slope of RS and TQ: -2