Let's solve the problem step-by-step to find the two different angle measures of the parallelogram-shaped tile.
1. Identify the given expressions for the angles:
- One angle is [tex]\( (6n - 70)^\circ \)[/tex].
- The opposite angle is [tex]\( (2n + 10)^\circ \)[/tex].
2. Recall the property of a parallelogram:
- Opposite angles of a parallelogram are equal.
- Adjacent angles of a parallelogram are supplementary (i.e., they add up to [tex]\(180^\circ\)[/tex]).
3. Set up the equation for supplementary angles:
- The sum of the given angles, [tex]\( (6n - 70)^\circ \)[/tex] and [tex]\( (2n + 10)^\circ \)[/tex], should be equal to [tex]\(180^\circ\)[/tex]:
[tex]\[
(6n - 70)^\circ + (2n + 10)^\circ = 180^\circ
\][/tex]
4. Combine like terms:
[tex]\[
6n - 70 + 2n + 10 = 180
\][/tex]
5. Simplify the equation:
[tex]\[
8n - 60 = 180
\][/tex]
6. Solve for [tex]\(n\)[/tex]:
[tex]\[
8n = 240
\][/tex]
[tex]\[
n = 30
\][/tex]
7. Substitute [tex]\(n\)[/tex] back into the angle expressions to find their measures:
- First angle: [tex]\(6n - 70\)[/tex]
[tex]\[
6(30) - 70 = 180 - 70 = 110^\circ
\][/tex]
- Second angle: [tex]\(2n + 10\)[/tex]
[tex]\[
2(30) + 10 = 60 + 10 = 70^\circ
\][/tex]
8. Determine the measures of all four angles in the parallelogram:
- Since in a parallelogram, opposite angles are equal, we have:
[tex]\[
(6n - 70)^\circ = 110^\circ
\][/tex]
[tex]\[
(2n + 10)^\circ = 70^\circ
\][/tex]
Therefore, the two different angle measures of the parallelogram-shaped tile are [tex]\(70^\circ\)[/tex] and [tex]\(110^\circ\)[/tex].
The correct answer is:
[tex]\[
\boxed{70^\circ \text{ and } 110^\circ}
\][/tex]
