To determine the measure of angle \( m\angle A \) in the inscribed quadrilateral \( MATH \), we can use the properties of cyclic quadrilaterals, which tell us that opposite angles are supplementary. This means that:
\[ m\angle A + m\angle C = 180^\circ \] \[ m\angle B + m\angle D = 180^\circ \]
Given the measures of the arcs:
- Arc \( AT \) is \( 50^\circ \)
- Arc \( TH \) is \( 170^\circ \)
Now, we can find the measures of angles \( A \), \( T \), \( H \), and \( M \) based on the intercepted arcs:
- The measure of angle \( A \) is half the measure of the arc opposite it, which is arc \( TH \).
- The measure of angle \( T \) is half the measure of the arc opposite it, which is arc \( MA \).
- The measure of angle \( H \) is half the measure of the arc opposite it, which is arc \( AT \).
- The measure of angle \( M \) is half the measure of the arc opposite it, which is arc \( MH \).
First, calculate the measures of angles:
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Finding \( m\angle H \): \[ m\angle H = \frac{1}{2} \times m\text{(arc opposite to H)} = \frac{1}{2} \times 50^\circ = 25^\circ \]
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Finding \( m\angle T \): Since \( m\angle T \) is opposite to arc \( MA \) (which we still need to determine), let's calculate \( m\angle A \) first.
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Calculating \( m\angle A \): We know the remaining angles based on the properties of cyclic quadrilaterals and that the opposite angles sum up to \(180^\circ\): \[ m\angle A + m\angle H = 180^\circ \] But we already know \( m\angle H = 25^\circ \): \[ m\angle A + 25^\circ = 180^\circ \quad \Rightarrow \quad m\angle A = 180^\circ - 25^\circ = 155^\circ \]
Therefore, the measure of angle \( m\angle A \) is: \[ \boxed{155^\circ} \]