Use the image to answer the question.

Question

Quadrilateral upper M upper A upper T upper H is drawn within a circle such that all vertices touch the edge of the circle. The arc joining upper A and upper T is labeled 50 degrees. The arc joining upper T and upper H is marked 170 degrees.

The image shows an inscribed quadrilateral. The mAM=54°
. Determine the m∠A
.

(1 point)
m∠A=
°

Answer 1

In an inscribed quadrilateral, opposite angles are supplementary. This means that the measures of the angles must add up to \(180^\circ\).

Given that \(m \angle M = 54^\circ\), we can find \(m \angle A\) as follows:

Since the arcs in the circle correspond to the angles opposite to them, we have:

1. The arc from \(A\) to \(T\) measures \(50^\circ\): This means the inscribed angle \(\angle M\) subtended by arc \(AT\) would measure half of the arc: \[ m \angle M = \frac{1}{2} \times 50^\circ = 25^\circ \]

2. The arc from \(T\) to \(H\) measures \(170^\circ\): Thus, the inscribed angle \(\angle A\) subtended by arc \(TH\) would measure: \[ m \angle A = \frac{1}{2} \times 170^\circ = 85^\circ \]

Now, recall that \(m \angle A + m \angle M = 180^\circ\). Since \(m \angle M\) was found through arc \(AT\), we actually want to consider the relationship correctly:

Using \(m \angle A + m \angle M = 180^\circ\), where \(m \angle M = 54^\circ\): \[ m \angle A = 180^\circ - 54^\circ = 126^\circ \]

Thus: \[ m \angle A = 126^\circ \]

So, the final answer is: \[ \boxed{126} \] degrees.