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The problem states that chords \( AB \) and \( CD \) are perpendicular and intersect at point \( E \) within the circle \( K \). We're given that \( m\angle BD = 126^\circ \) and asked to find the measure of arc \( AC \).

### Key points:
1. The perpendicular intersection of the two chords suggests that point \( E \) is important for symmetry or special geometric relationships.
2. \( m\overarc{BD} = 126^\circ \), which means the arc subtended by chord \( BD \) measures \( 126^\circ \).
3. To find \( m\overarc{AC} \), we will use properties of the circle, specifically the fact that the opposite arcs in a circle are supplementary. Since the full circle is \( 360^\circ \), the arc \( BD \) and the arc \( AC \) together must add up to \( 360^\circ \).

### Solution steps:
- The total angle around the circle is \( 360^\circ \). Since \( m\overarc{BD} = 126^\circ \), the remaining arc, which is \( m\overarc{AC} \), can be calculated as:
  \[
  m\overarc{AC} = 360^\circ - 126^\circ = 234^\circ
  \]

Thus, the measure of arc \( AC \) is \( 234^\circ \).

Would you like further details or explanations on this? 

Here are 5 related questions you might find helpful:
1. How do perpendicular chords influence arc measures in a circle?
2. What are the properties of intersecting chords in a circle?
3. How do you determine the measure of a central angle subtended by an arc?
4. How do supplementary arcs work in a circle?
5. Can this method be used for any intersecting chords in a circle?

**Tip:** Always remember that opposite arcs in a circle are supplementary, which can help in solving many circle-related problems.

In the diagram of circle K, chords AB and CD are perpendicular and intersect at E. If mBD = 126°, then determine mAC.

This problem involves finding the measure of arc AC in a circle where two perpendicular chords intersect at a point. Given that mBD is 126°, use the supplementary arcs theorem to find mAC. Suitable for high school students in Grades 9-12.

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