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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]
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]
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