To find the two different angle measures of the parallelogram-shaped tile, we first need to address the relationship between the given angles and use the properties of a parallelogram.
Given two opposite angles with expressions:
[tex]\[ \text{Angle 1} = (6n - 70)^\circ \][/tex]
[tex]\[ \text{Angle 2} = (2n + 10)^\circ \][/tex]
We know that opposite angles in a parallelogram are equal. Therefore,
[tex]\[ 6n - 70 = 2n + 10 \][/tex]
To solve for \( n \), follow these steps:
1. Set up the equation:
[tex]\[ 6n - 70 = 2n + 10 \][/tex]
2. Isolate \( n \):
Subtract \( 2n \) from both sides:
[tex]\[ 6n - 2n - 70 = 2n - 2n + 10 \][/tex]
[tex]\[ 4n - 70 = 10 \][/tex]
Now, add 70 to both sides:
[tex]\[ 4n - 70 + 70 = 10 + 70 \][/tex]
[tex]\[ 4n = 80 \][/tex]
3. Solve for \( n \):
[tex]\[ n = \frac{80}{4} \][/tex]
[tex]\[ n = 20 \][/tex]
Next, substitute \( n = 20 \) back into the expressions for the angles:
4. Calculate the measures:
[tex]\[ \text{Angle 1} = 6n - 70 = 6(20) - 70 = 120 - 70 = 50^\circ \][/tex]
[tex]\[ \text{Angle 2} = 2n + 10 = 2(20) + 10 = 40 + 10 = 50^\circ \][/tex]
Both calculated angles (opposite angles in the parallelogram) are equal to \( 50^\circ \).
In a parallelogram, the adjacent angles are supplementary (they add up to \( 180^\circ \)). Therefore, to find the measure of the adjacent angle, we subtract the given angle from \( 180^\circ \):
5. Calculate the adjacent angles:
[tex]\[ \text{Adjacent Angle to 50} = 180^\circ - 50^\circ = 130^\circ \][/tex]
Thus, the two different angle measures in the parallelogram are \( 50^\circ \) and \( 130^\circ \).
The correct option is:
[tex]\[ 50^\circ \text{ and } 130^\circ \][/tex]
