To determine the two different angle measures of the parallelogram-shaped tile, we need to utilize some properties of parallelograms and solve for [tex]\( n \)[/tex].
### Step-by-Step Solution:
1. Understand the Properties of Angles in a Parallelogram:
- Opposite angles in a parallelogram are equal.
- Adjacent angles in a parallelogram are supplementary, meaning that the sum of the measures of two adjacent angles is [tex]\(180^\circ\)[/tex].
2. Set Up the Equations:
- We know the measures of the two given angles: [tex]\( (6n - 70)^\circ \)[/tex] and [tex]\( (2n + 10)^\circ \)[/tex].
- According to the supplementary property of adjacent angles in a parallelogram:
[tex]\[
(6n - 70) + (2n + 10) = 180^\circ
\][/tex]
3. Simplify and Solve the Equation:
- Combine like terms:
[tex]\[
6n - 70 + 2n + 10 = 180
\][/tex]
- This simplifies to:
[tex]\[
8n - 60 = 180
\][/tex]
- To isolate [tex]\( n \)[/tex], add 60 to both sides of the equation:
[tex]\[
8n = 240
\][/tex]
- Divide both sides by 8 to solve for [tex]\( n \)[/tex]:
[tex]\[
n = 30
\][/tex]
4. Calculate the Measures of the Angles:
- Substitute [tex]\( n = 30 \)[/tex] back into the expressions for the angles:
[tex]\[
(6n - 70)^\circ = 6(30) - 70 = 180 - 70 = 110^\circ
\][/tex]
[tex]\[
(2n + 10)^\circ = 2(30) + 10 = 60 + 10 = 70^\circ
\][/tex]
5. Verify the Angles:
- Check that opposite angles are equal:
- The expressions [tex]\( (6n - 70)^\circ \)[/tex] and [tex]\( (2n + 10)^\circ \)[/tex] when evaluated for [tex]\( n = 30 \)[/tex], give us [tex]\( 110^\circ \)[/tex] and [tex]\( 70^\circ \)[/tex] respectively.
- Check the supplementary property:
- Confirm that [tex]\( 110^\circ + 70^\circ = 180^\circ \)[/tex], which is correct.
### Conclusion:
The two different angle measures of the parallelogram-shaped tile are [tex]\( 70^\circ \)[/tex] and [tex]\( 110^\circ \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{70^\circ \text{ and } 110^\circ}
\][/tex]
