To make a conjecture about \( m\angle A \) based on the given angle measures for quadrilateral \( ABDE \):
- The sum of the angles in any quadrilateral is \( 360^\circ \).
- We know the angles \( m\angle B = 40^\circ \), \( m\angle D = 60^\circ \), and \( m\angle E = 80^\circ \).
- By adding these angles together, we find the measure of angle \( A \):
\[ m\angle A = 360^\circ - (m\angle B + m\angle D + m\angle E) \]
Calculating the sum:
\[ m\angle B + m\angle D + m\angle E = 40^\circ + 60^\circ + 80^\circ = 180^\circ \]
Now, substitute this sum into the equation for \( m\angle A \):
\[ m\angle A = 360^\circ - 180^\circ = 180^\circ \]
However, angle \( A \) is part of triangle \( AABC \), not the quadrilateral \( ABDE \). If considering triangles and maintains corresponding angle patterns, it could change based on their relationship in the figure or similar triangles.
Assuming the angles \( m\angle AAB \) and corresponding relationships derive \( A\), we reconsider its options and relationships:
Given the options provided consider only the angle patterns derived from existing for a potential triangle or dependent relation.
Thus, since A derives itself from relationships, moving into constructed triangle through A gives Option # 2: m<A = 40 Degrees as likely true since a direct derivation comes through angle relationships normalized for similar behavior as provided by angles existing, thus the only compatible relationship and conjecture check derived angles being:
Final answer: Option # 2: \( m\angle A = 40^\circ \).