To solve for \( x \) with the given angles \( m\angle 1 = 14x^\circ \) and \( m\angle 2 = 2 - \frac{x}{5} \) degrees, we'll first write the equation based on the relationship between these angles.
(a) Write an Equation
Since the problem does not specify whether the angles are equal or some other relationship, I'll assume that \( m\angle 1 \) and \( m\angle 2 \) are equal. Thus, we can write:
\[ 14x = 2 - \frac{x}{5} \]
(b) Solve for \( x \) and Find Each Angle
To solve for \( x \):
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Eliminate the fraction by multiplying every term by 5 to simplify: \[ 5(14x) = 5\left(2 - \frac{x}{5}\right) \] This results in: \[ 70x = 10 - x \]
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Now, add \( x \) to both sides: \[ 70x + x = 10 \] This simplifies to: \[ 71x = 10 \]
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Now, divide both sides by 71: \[ x = \frac{10}{71} \]
Now we can find the degree measure of each angle.
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For \( m\angle 1 \): \[ m\angle 1 = 14x = 14 \times \frac{10}{71} = \frac{140}{71} \approx 1.97^\circ \]
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For \( m\angle 2 \): \[ m\angle 2 = 2 - \frac{x}{5} \] First calculate \( \frac{x}{5} \): \[ \frac{x}{5} = \frac{10/71}{5} = \frac{10}{355} = \frac{2}{71} \] Thus, \[ m\angle 2 = 2 - \frac{2}{71} = \frac{142}{71} - \frac{2}{71} = \frac{140}{71} \approx 1.97^\circ \]
Final Answers
Thus, we have:
- Equation: \( 14x = 2 - \frac{x}{5} \)
- Degree measures: \[ m\angle 1 \approx 1.97^\circ \] \[ m\angle 2 \approx 1.97^\circ \]
Both angles will have the same measure, as expected from our equation.