To find the measure \( m\angle KIJ \), we first need to understand that the measure of an angle formed by two intersecting chords or arcs can be determined using the formula:
\[ m\angle KIJ = \frac{1}{2} (m\text{arc KJ} + m\text{arc JI}) \]
You provided the measures of arc KJ and arc JI as follows:
- \( m\text{arc KJ} = 13x - 10 \)
- \( m\text{arc JI} = 7x - 10 \)
Now substitute these into the formula for \( m\angle KIJ \):
\[ m\angle KIJ = \frac{1}{2} \left( (13x - 10) + (7x - 10) \right) \]
Combine the terms inside the parentheses:
\[ m\angle KIJ = \frac{1}{2} \left( 20x - 20 \right) \]
Now simplify further:
\[ m\angle KIJ = \frac{20x - 20}{2} = 10x - 10 \]
Now we need to set \( 10x - 10 \) equal to the possible angle measures you provided (120, 80, 60, and 10 degrees) to find the value of \( x \) for each case.
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For \( m\angle KIJ = 120 \): \[ 10x - 10 = 120 \] \[ 10x = 130 \quad \Rightarrow \quad x = 13 \]
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For \( m\angle KIJ = 80 \): \[ 10x - 10 = 80 \] \[ 10x = 90 \quad \Rightarrow \quad x = 9 \]
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For \( m\angle KIJ = 60 \): \[ 10x - 10 = 60 \] \[ 10x = 70 \quad \Rightarrow \quad x = 7 \]
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For \( m\angle KIJ = 10 \): \[ 10x - 10 = 10 \] \[ 10x = 20 \quad \Rightarrow \quad x = 2 \]
Now we can conclude that \( m<KIJ \) could be any of the angles depending on the value of \( x \). Now check if any of the values for \( x \) make both arcs positive.
For \( x = 13 \):
- Arc KJ = \( 13(13) - 10 = 169 - 10 = 159 \)
- Arc JI = \( 7(13) - 10 = 91 - 10 = 81 \)
Both arcs are positive.
For \( x = 9 \):
- Arc KJ = \( 13(9) - 10 = 117 - 10 = 107 \)
- Arc JI = \( 7(9) - 10 = 63 - 10 = 53 \)
Both arcs are positive.
For \( x = 7 \):
- Arc KJ = \( 13(7) - 10 = 91 - 10 = 81 \)
- Arc JI = \( 7(7) - 10 = 49 - 10 = 39 \)
Both arcs are positive.
For \( x = 2 \):
- Arc KJ = \( 13(2) - 10 = 26 - 10 = 16 \)
- Arc JI = \( 7(2) - 10 = 14 - 10 = 4 \)
Both arcs are positive.
Thus, all angle measures are feasible depending on the value of \( x \).
The options provided that fits one of those values is 80 degrees, so the most straightforward and correct response is:
80 degrees.