To determine the expression that represents the measure of angle \( m \) in triangle \( ABC \), given that line segment \( BM \) bisects angle \( ABC \) and line segment \( BK \) is the height of triangle \( ABC \), we can apply some geometric properties.
Assuming that:
- \( \angle ABC \) is denoted as \( \theta \).
- Since \( BM \) bisects \( \angle ABC \), we have \( \angle ABM = \angle MBC = \frac{\theta}{2} \).
The height \( BK \) from point \( B \) to line \( AC \) is perpendicular to \( AC \), which means:
- Angle \( BKA \) (or angle \( BKC \)) is \( 90^\circ \).
If \( m \) refers to one of the angles formed at vertex \( B \) considering the bisector and the height, we could describe \( \angle ABK \) using these properties.
Given that \( BK \) is perpendicular to \( AC \):
- \( \angle ABK = \angle ABM + \angle MBK = \frac{\theta}{2} + 90^\circ \).
If \( m \) refers to the angle \( ABK \):
\[ m = \frac{\theta}{2} + 90^\circ, \]
where \( \theta \) is the measure of angle \( ABC \).
Thus, if you need to determine a specific expression for \( m \) in terms of \( \theta \) or some other defined angles in triangle \( ABC \), you may need to provide additional details or functional relationships. But typically, in reference to the properties of bisectors and heights, that would be the involved expressions.
If you have a specific angle or dimension to be defined symbolically, please provide that, and I’d be happy to help further!