To solve for \( m∠2 \), we can use the information given in the problem. We know that angles 1 and 2 together form a right angle, meaning:
\[ m∠1 + m∠2 = 90° \]
It is also given that \( m∠4 = 31° \). By the configuration of the rays (assuming they are in a straight line), we know that:
\[ m∠3 + m∠4 + m∠5 = 180° \]
Since angles 4 and 5 are adjacent to angle 3, this means:
\[ m∠3 + 31° + m∠2 = 180° \]
Now, since angles 2 and 3 are also part of a linear relationship with angles 1 and 4, we can formulate another relationship. Knowing that angle 1 is the complement of angle 2:
\[ m∠3 + m∠1 + m∠2 = 180° \]
Combining these facts, we can solve \( m∠2 \):
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From \( m∠1 + m∠2 = 90° \), we can express \( m∠1 = 90° - m∠2 \).
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Substitute \( m∠1 \) into the linear equation involving angle 3:
\[ m∠3 + (90° - m∠2) + m∠2 = 180° \]
This simplifies to:
\[ m∠3 + 90° = 180° \]
From which we find:
\[ m∠3 = 90° \]
Thus, with \( m∠4 = 31° \):
\[ m∠3 + m∠4 + m∠5 = 180° \] \[ 90° + 31° + m∠5 = 180° \] \[ m∠5 = 180° - 121° = 59° \]
Now, we also know
\[ m∠4 + m∠5 = 31° + 59° = 90° \]
Thus:
\[ m∠1 + m∠2 + m∠3 = 180° \]
Keeping in mind that \( m∠1 + m∠2 = 90°\) gives us the value of \( m∠2 \), since angles add up linearly around the point where these rays emerge.
Since \( m∠2 \) is supplementary to the \( m∠1 \):
If you take \( m∠4 = 31°\), you can justify
Therefore, \( m∠2 = 59° \).
So the answer is:
59°.
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