To find the value of \( x \) and the measures of the two angles, we start with the information given in the problem.
The two angles are:
- \( m∠1 = 3x° \)
- \( m∠2 = (2x + 10)° \)
According to the problem, these two angles are adjacent and form a resulting angle of \( 160° \). Therefore, we can set up the equation:
\[ m∠1 + m∠2 = 160° \]
Substituting the expressions for \( m∠1 \) and \( m∠2 \):
\[ 3x + (2x + 10) = 160 \]
Now, simplify this equation:
\[ 3x + 2x + 10 = 160 \] \[ 5x + 10 = 160 \]
Next, isolate \( x \) by first subtracting \( 10 \) from both sides:
\[ 5x = 160 - 10 \] \[ 5x = 150 \]
Now, divide by \( 5 \):
\[ x = \frac{150}{5} \] \[ x = 30 \]
Now that we have \( x = 30 \), we can find the measures of the angles:
For \( m∠1 \):
\[ m∠1 = 3x = 3(30) = 90° \]
For \( m∠2 \):
\[ m∠2 = 2x + 10 = 2(30) + 10 = 60 + 10 = 70° \]
Now we confirm if the sum is \( 160° \):
\[ m∠1 + m∠2 = 90° + 70° = 160° \]
This verification is correct. Thus, the measures of the angles are:
- \( m∠1 = 90° \)
- \( m∠2 = 70° \)
So the final answer is:
- \( m∠1 = 90° \)
- \( m∠2 = 70° \)
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