To find the measure of the arc BCD in circle A, we first need to understand the relationship between the angles inscribed in the circle and the arcs they subtend.
Given arc BCD, we know that the measure of an arc in a circle is equal to the measure of the angle that subtends it at the center of the circle. Since the angle that subtends the arc BCD from point A is not provided directly, we consider the relevant angles provided with respect to the circle.
- From your question formatting, it looks like you have angles of 58°, 90°, and 57° mentioned, but it’s not clear which angles are inscribed angles or central angles.
- Typically, the inscribed angle theorem states that the measure of an inscribed angle is half the measure of the arc that it intercepts.
Assuming:
- m∠BCA = 58° (an inscribed angle)
- The opposite angle that subtends arc BCD is ∠BAC or ∠CAD (which will add to angles forming a full circle).
If we are to assume that both BCA and the angle opposite subtending BCD are around the circle in such a manner that they contribute to arc BCD, we typically sum up to find the angle subtending BCD.
Using the given angles:
- m∠ABC (if it's adjacent to supplied angle): This can be combined with other angles (assuming it completes circles with provided angles) to solve for BCD.
- To find mArc BCD:
- Often given \( m∠BCA \) might relate to BCD in the way of arcs (58° would help calculate how much spans on the circle).
- If the angle BCA is part of a vertex controlled by two of the other provided angles, knowing these might inspire calculation for total obtuse or acute angles leading to full 360 degrees.
Assuming we finally find how these angles correspond to form mArc BCD ranging effectively from properties of complementary or supplementary angles, you might arrive at the arc measure through linear or even direct angle addition depending on configuration.
Conclusion:
If you can clarify or give configuration (for example, “if BCD subtends specific angles at A”), it's possible to obtain direct numerical values fitting \( mArc BCD = 90 - x \) if forming a circle and subsets as suggested by the measures provided.
The final answer would ideally be contextualized based on segment complements in arc measures and would be calculated as thus.
Please clarify angle assignments for precise calculation of \( mArc BCD \).