To find the two different angle measures of the parallelogram-shaped tile, let's go through the problem step-by-step.
Given that two opposite angles of the parallelogram are [tex]\( (6n - 70)^\circ \)[/tex] and [tex]\( (2n + 10)^\circ \)[/tex], we know that these angles must be equal because opposite angles in a parallelogram are always equal.
1. Set the angles equal to each other:
[tex]\[
6n - 70 = 2n + 10
\][/tex]
2. Solve for [tex]\( n \)[/tex]:
[tex]\[
6n - 2n = 10 + 70
\][/tex]
Simplify the equation:
[tex]\[
4n = 80
\][/tex]
Divide by 4:
[tex]\[
n = 20
\][/tex]
3. Calculate the angles using [tex]\( n = 20 \)[/tex]:
[tex]\[
\text{First angle} = 6n - 70 = 6(20) - 70 = 120 - 70 = 50^\circ
\][/tex]
[tex]\[
\text{Second angle} = 2n + 10 = 2(20) + 10 = 40 + 10 = 50^\circ
\][/tex]
4. Since opposite angles in a parallelogram are equal, we have:
[tex]\[
\text{Angle 1} = \text{Angle 2} = 50^\circ
\][/tex]
5. In a parallelogram, consecutive angles are supplementary (they add up to [tex]\(180^\circ\)[/tex]). Therefore, the other two angles are:
[tex]\[
\text{Angle 3} = 180^\circ - 50^\circ = 130^\circ
\][/tex]
[tex]\[
\text{Angle 4} = 180^\circ - 50^\circ = 130^\circ
\][/tex]
6. Conclusion:
The two different angle measures of the parallelogram-shaped tile are [tex]\( 50^\circ \)[/tex] and [tex]\( 130^\circ \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{50^\circ \text{ and } 130^\circ}
\][/tex]
