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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]


Sagot :

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]