To solve the problem, we will use the properties of angles formed by intersecting lines. Given:

- \( aHc = 60^\circ \)
- \( bHm = 44^\circ \)

Let's find the required angles step by step.

### a) Calculate \( \hat{H_1}, \hat{H_2} \)

From the given angle \( aHc = 60^\circ \):
- The angle on the opposite side, \( \hat{H_1} = \hat{aHb} \), is equal to \( 60^\circ \).

From angle \( bHm = 44^\circ \):
- The angle on the opposite side, \( \hat{H_2} = \hat{bHd} \), is equal to \( 44^\circ \).

### b) Calculate \( bHc, bHn \)

Now, we know:
- \( \hat{bHm} = 44^\circ \) and hence, \( bHc + bHm = 180^\circ \) because they are supplementary angles (same side of the transversal).

Thus,
\[
bHc = 180^\circ - 44^\circ = 136^\circ
\]

Now for \( bHn \):
- Using \( \hat{H_1} = 60^\circ \):
\[
bHn + bHc = 180^\circ
\]
So,
\[
bHn = 180^\circ - 136^\circ = 44^\circ
\]

### c) Calculate \( mHc \)

Using \( aHc = 60^\circ \):
- \( mHc \) is vertically opposite to \( bHm \), thus:
\[
mHc = bHm = 44^\circ
\]

### Summary of Results

a) \( \hat{H_1} = 60^\circ, \hat{H_2} = 44^\circ \)

b) \( bHc = 136^\circ, bHn = 44^\circ \)

c) \( mHc = 44^\circ \)