To determine the two different angle measures of Jacob's parallelogram-shaped tile, we start by using the given angle expressions:
1. One of the angles is [tex]\(6n - 70\)[/tex] degrees.
2. The opposite angle is [tex]\(2n + 10\)[/tex] degrees.
Since opposite angles in a parallelogram are equal, we can set the two expressions equal to each other and solve for [tex]\(n\)[/tex]:
[tex]\[
6n - 70 = 2n + 10
\][/tex]
First, we need to isolate [tex]\(n\)[/tex] on one side of the equation. To do this, we subtract [tex]\(2n\)[/tex] from both sides:
[tex]\[
6n - 2n - 70 = 10
\][/tex]
Simplify the equation:
[tex]\[
4n - 70 = 10
\][/tex]
Next, we add 70 to both sides to isolate the term with [tex]\(n\)[/tex]:
[tex]\[
4n = 80
\][/tex]
Now, we divide both sides by 4 to solve for [tex]\(n\)[/tex]:
[tex]\[
n = 20
\][/tex]
With [tex]\(n = 20\)[/tex], we substitute back into the expressions for the angles to find the specific measures:
For the first angle:
[tex]\[
6n - 70 = 6(20) - 70 = 120 - 70 = 50^\circ
\][/tex]
For the opposite angle:
[tex]\[
2n + 10 = 2(20) + 10 = 40 + 10 = 50^\circ
\][/tex]
Then, determine the measure of the other pair of angles in the parallelogram. Since the angles in any quadrilateral sum to [tex]\(360^\circ\)[/tex], and a parallelogram has two pairs of equal opposite angles, we calculate the remaining angles:
[tex]\[
180^\circ - 50^\circ = 130^\circ
\][/tex]
Thus, the two different angle measures of the parallelogram-shaped tile are:
[tex]\[
50^\circ \text{ and } 130^\circ
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{50^\circ \text{ and } 130^\circ}
\][/tex]
