To determine the values of \( m \) and \( n \) that guarantee ABCD is a parallelogram, we will use the property that in a parallelogram, opposite angles are equal and the sum of the angles in any quadrilateral is \( 360^\circ \).
Given: \[ A = 3n + 3 \] \[ B = 5m \] \[ C = 3m + 30 \] \[ D = 105 \]
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Using the property of opposite angles: For quadrilateral ABCD to be a parallelogram: \[ A + C = B + D \]
Substituting the angles: \[ (3n + 3) + (3m + 30) = (5m) + 105 \]
Simplifying this: \[ 3n + 3 + 3m + 30 = 5m + 105 \] \[ 3n + 33 = 5m + 105 \] \[ 3n = 5m + 105 - 33 \] \[ 3n = 5m + 72 \] \[ n = \frac{5m + 72}{3} \]
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Using the property of the sum of angles: The sum of the angles in quadrilateral ABCD should equal \( 360^\circ \): \[ A + B + C + D = 360 \]
Substituting the angles: \[ (3n + 3) + (5m) + (3m + 30) + (105) = 360 \]
Simplifying this: \[ 3n + 3 + 5m + 3m + 30 + 105 = 360 \] \[ 3n + 8m + 138 = 360 \] \[ 3n + 8m = 360 - 138 \] \[ 3n + 8m = 222 \] \[ n = \frac{222 - 8m}{3} \]
Now we have two equations:
- \( n = \frac{5m + 72}{3} \) (from opposite angles)
- \( n = \frac{222 - 8m}{3} \) (from sum of angles)
Setting these two expressions for \( n \) equal to each other: \[ \frac{5m + 72}{3} = \frac{222 - 8m}{3} \]
Since they both have a common denominator, we can eliminate it: \[ 5m + 72 = 222 - 8m \]
Now, solving for \( m \): \[ 5m + 8m = 222 - 72 \] \[ 13m = 150 \] \[ m = \frac{150}{13} \] \[ m \approx 11.54 \]
Now substituting back to find \( n \): \[ n = \frac{5\left(\frac{150}{13}\right) + 72}{3} \] \[ n = \frac{\frac{750}{13} + \frac{936}{13}}{3} \] \[ n = \frac{\frac{1686}{13}}{3} = \frac{1686}{39} \] \[ n = \frac{562}{13} \] \[ n \approx 43.23 \]
Thus, the values of \( m \) and \( n \) that guarantee that ABCD is a parallelogram are: \[ m = \frac{150}{13}, \quad n = \frac{562}{13} \] or numerically approximately \( m \approx 11.54 \) and \( n \approx 43.23 \).
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