To find the measures of the two unknown angles in the quadrilateral, we first need to understand that the sum of the angles in a quadrilateral is always \(360^\circ\).
We are given two angles:
- Angle 1: \(277^\circ\)
- Angle 2: \(53^\circ\)
Now we can calculate the sum of these two angles: \[ 277^\circ + 53^\circ = 330^\circ \]
Next, we can determine the sum of the remaining two angles: \[ 360^\circ - 330^\circ = 30^\circ \]
Let the measures of the two remaining angles be \(7x\) and \(8x\) as they are in a ratio of \(7:8\).
Now, we can set up the equation based on their sum: \[ 7x + 8x = 30^\circ \] \[ 15x = 30^\circ \]
Now, solving for \(x\): \[ x = \frac{30^\circ}{15} = 2^\circ \]
Now we can find the measures of the two angles: \[ 7x = 7 \times 2^\circ = 14^\circ \] \[ 8x = 8 \times 2^\circ = 16^\circ \]
Thus, the measures of the two angles are: \[ \boxed{14^\circ} \quad \text{and} \quad \boxed{16^\circ} \]
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