To solve for \( m∠2 \), we first note that angles \( m∠1 \) and \( m∠2 \) form a right angle, meaning:
\[ m∠1 + m∠2 = 90° \]
Given that \( m∠4 = 31° \), we can analyze the relationships between the angles. Since the rays are emerging from the same point, we also have the following relationships:
- \( m∠3 \) is supplementary to \( m∠4 \) and \( m∠5 \).
Assuming the angles are arranged sequentially in a circle:
- \( m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360° \)
From the right angle information, let's express angle \( m∠5 \):
Since \( m∠3 = m∠1 + m∠2 \) (as they are adjacent), that means: \[ m∠3 = 90° - m∠2 \]
Now substituting: \[ (90° - m∠2) + m∠4 + m∠5 = 360° \]
Substituting \( m∠4 = 31° \): \[ (90° - m∠2) + 31° + m∠5 = 360° \] \[ 121° - m∠2 + m∠5 = 360° \] \[ m∠5 = 360° - 121° + m∠2 \] \[ m∠5 = 239° + m∠2 \]
From this, if we return to our earlier equation from \( m∠1 + m∠2 = 90° \), we can find \( m∠2 \) by substituting values and knowing that \( m∠4 = 31°\) guides us in measuring angle relationships.
Given that: \[ m∠2 = 90° - m∠1 \] Also,
If we sum up to \( m∠4 = 31° \): \[ m∠1 + m∠2 + m∠4 + m∠5 + m∠4 = 180° (half circle) \]
However, let's calculate the remaining angles based on simple scenarios:
- With \( m∠1 + m∠2 = 90°\) and they are basic part.
Thus we can solve directly for angle positioning:
\[ \text{Assuming that} \ m∠1 = 59° \to \text{thus it gives} \ m∠2 = 90° - 59° \Rightarrow m∠2 = 31° \]
Hence, evaluating through choices provided looks directly to maintain:
Final Answer: \( m∠2 = 59° \).
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