Question

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 meausre of angle R is (108-2x).
Determine if quadriateral QRST is a parallelogram, and if so, solve for x.

Answer 1

To determine whether quadrilateral QRST is a parallelogram, we can use the following properties of a parallelogram:

1. Opposite angles are equal.
2. The sum of the angles in any quadrilateral is \(360^\circ\).

Let's first calculate the sum of the angles in quadrilateral QRST. We know:

- \(m\angle Q = 108^\circ\)
- \(m\angle R = 108 - 2x\)

Let \(m\angle S\) and \(m\angle T\) be the measures of the remaining angles. We do not have their measures, but we can express the sum of the angles in QRST as follows:

\[
m\angle Q + m\angle R + m\angle S + m\angle T = 360^\circ
\]

Substituting the known values, we have:

\[
108 + (108 - 2x) + m\angle S + m\angle T = 360
\]

Simplifying this equation gives:

\[
216 - 2x + m\angle S + m\angle T = 360
\]

Now, combining terms:

\[
m\angle S + m\angle T = 360 - 216 + 2x
\]
\[
m\angle S + m\angle T = 144 + 2x
\]

In a parallelogram, opposite angles are equal. So, if we assume that angles \(Q\) and \(S\) are opposite, and angles \(R\) and \(T\) are opposite, we conclude:

1. \(m\angle Q = m\angle S\)
2. \(m\angle R = m\angle T\)

From the first equation, we have:

\[
m\angle S = 108^\circ
\]

From the second equation, we can express angle \(T\):

\[
m\angle T = 108 - 2x
\]

Now, setting the two expressions for \(m\angle R\) and \(m\angle T\) equal, we have:

\[
108 - 2x = m\angle T
\]

We know that:

\[
m\angle S + m\angle T = 144 + 2x
\]

Substituting \(m\angle S = 108^\circ\):

\[
108 + m\angle T = 144 + 2x
\]

This simplifies to:

\[
m\angle T = 144 + 2x - 108
\]
\[
m\angle T = 36 + 2x
\]

Now, we can set \(m\angle T\) equal to our earlier expression \(108 - 2x\):

\[
36 + 2x = 108 - 2x
\]

Next, we combine like terms:

\[
2x + 2x = 108 - 36
\]
\[
4x = 72
\]

Dividing both sides by 4 gives:

\[
x = 18
\]

### Conclusion
To determine if quadrilateral QRST is a parallelogram, we can verify:

1. If \(x = 18\), then:
- \(m\angle R = 108 - 2(18) = 108 - 36 = 72^\circ\)
- \(m\angle T = 108 - 2(18) = 72^\circ\)

Since \(m\angle S = 108^\circ\) and \(m\angle Q = 108^\circ\), and \(m\angle R = m\angle T = 72^\circ\), we see that opposite angles are equal, confirming that QRST is indeed a parallelogram.

Thus, the solution is:

- Quadrilateral QRST is a parallelogram, and \(x = 18\).