To determine which angle in quadrilateral JKLM is the greatest, we need to use the fact that JKLM is inscribed in a circle. For any quadrilateral inscribed in a circle, the opposite angles are supplementary, meaning their measures add up to [tex]\(180^\circ\)[/tex].
First, note the given information:
- [tex]\( m\angle 3 = 95^\circ \)[/tex]
- [tex]\( m\angle K = (215-4x)^\circ \)[/tex]
- [tex]\( m\angle L = (2x+15)^\circ \)[/tex]
- [tex]\( m\angle M = (140-x)^\circ \)[/tex]
Since [tex]\( \angle K \)[/tex] is opposite [tex]\( \angle 3 \)[/tex], we have:
[tex]\[ m\angle K + m\angle 3 = 180^\circ \][/tex]
Substitute [tex]\(m\angle K\)[/tex] and [tex]\(m\angle 3\)[/tex]:
[tex]\[ (215-4x) + 95 = 180 \][/tex]
[tex]\[ 310 - 4x = 180 \][/tex]
Subtract 180 from both sides:
[tex]\[ 310 - 180 = 4x \][/tex]
[tex]\[ 130 = 4x \][/tex]
Divide both sides by 4:
[tex]\[ x = 32.5 \][/tex]
Next, we'll use this value of [tex]\(x\)[/tex] to find the measures of [tex]\( \angle K \)[/tex], [tex]\( \angle L \)[/tex], and [tex]\( \angle M \)[/tex]:
1. Calculate [tex]\( m\angle K \)[/tex]:
[tex]\[ m\angle K = 215 - 4 \cdot 32.5 \][/tex]
[tex]\[ m\angle K = 215 - 130 \][/tex]
[tex]\[ m\angle K = 85^\circ \][/tex]
2. Calculate [tex]\( m\angle L \)[/tex]:
[tex]\[ m\angle L = 2 \cdot 32.5 + 15 \][/tex]
[tex]\[ m\angle L = 65 + 15 \][/tex]
[tex]\[ m\angle L = 80^\circ \][/tex]
3. Calculate [tex]\( m\angle M \)[/tex]:
[tex]\[ m\angle M = 140 - 32.5 \][/tex]
[tex]\[ m\angle M = 107.5^\circ \][/tex]
Now we compare the measures of all angles:
- [tex]\( m\angle 3 = 95^\circ \)[/tex]
- [tex]\( m\angle K = 85^\circ \)[/tex]
- [tex]\( m\angle L = 80^\circ \)[/tex]
- [tex]\( m\angle M = 107.5^\circ \)[/tex]
The angle with the greatest measure is [tex]\( \angle M \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
