Jacob is cutting a tile in the shape of a parallelogram. Two opposite angles have measures of [tex]\((6n - 70)^{\circ}\)[/tex] and [tex]\((2n + 10)^{\circ}\)[/tex].

What are the two different angle measures of the parallelogram-shaped tile?

A. [tex]\(20^{\circ}\)[/tex] and [tex]\(160^{\circ}\)[/tex]
B. [tex]\(50^{\circ}\)[/tex] and [tex]\(130^{\circ}\)[/tex]
C. [tex]\(30^{\circ}\)[/tex] and [tex]\(150^{\circ}\)[/tex]
D. [tex]\(70^{\circ}\)[/tex] and [tex]\(110^{\circ}\)[/tex]



Answer :

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]