To find the measures of the two different angles of the parallelogram-shaped tile, let's start by noting an important property of parallelograms: opposite angles are equal and the sum of adjacent angles is [tex]\(180^{\circ}\)[/tex].
Given:
1. One pair of opposite angles are [tex]\( (6n - 70)^{\circ} \)[/tex]
2. The other pair of opposite angles are [tex]\( (2n + 10)^{\circ} \)[/tex]
Because opposite angles in a parallelogram are equal, we can set up the equation:
[tex]\[ 6n - 70 = 2n + 10 \][/tex]
Now, solve this equation for [tex]\( n \)[/tex]:
[tex]\[
6n - 2n = 70 + 10
\][/tex]
[tex]\[
4n = 80
\][/tex]
[tex]\[
n = 20
\][/tex]
Now, substitute [tex]\( n = 20 \)[/tex] back into the expressions for the angles to find the measures:
[tex]\[ 6n - 70 = 6(20) - 70 = 120 - 70 = 50 \][/tex]
[tex]\[ 2n + 10 = 2(20) + 10 = 40 + 10 = 50 \][/tex]
So one pair of opposite angles measures [tex]\( 50^{\circ} \)[/tex].
To find the adjacent angles, we use the fact that the sum of adjacent angles in a parallelogram is [tex]\( 180^{\circ} \)[/tex]:
[tex]\[ 180^{\circ} - 50^{\circ} = 130^{\circ} \][/tex]
Therefore, the two different angle measures of the parallelogram-shaped tile are:
[tex]\[ 50^{\circ} \text{ and } 130^{\circ} \][/tex]
So the correct answer is:
[tex]\[ \boxed{50^{\circ} \text{ and } 130^{\circ}} \][/tex]
