To determine which angle in the quadrilateral JKLM has the greatest measure, we need to solve the given expressions for the angles. Here is a systematic step-by-step approach:
1. Sum of the angles in a quadrilateral inscribed in a circle:
The sum of the interior angles of any quadrilateral is always [tex]\(360^{\circ}\)[/tex].
2. Given angles:
- [tex]\( m \angle J = 95^{\circ} \)[/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]
3. Equation setup:
The sum of the angles of quadrilateral JKLM is:
[tex]\[
m \angle J + m \angle K + m \angle L + m \angle M = 360^{\circ}
\][/tex]
Substituting the given values:
[tex]\[
95 + (215 - 4x) + (2x + 15) + (140 - x) = 360
\][/tex]
4. Simplifying the equation:
Combine like terms:
[tex]\[
95 + 215 + 15 + 140 - 4x + 2x - x = 360
\][/tex]
Combine all the constants:
[tex]\[
465 - 3x = 360
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Isolate the variable [tex]\(x\)[/tex]:
[tex]\[
465 - 360 = 3x
\][/tex]
[tex]\[
105 = 3x
\][/tex]
[tex]\[
x = 35
\][/tex]
6. Determine the measures of angles [tex]\(K\)[/tex], [tex]\(L\)[/tex], and [tex]\(M\)[/tex] using [tex]\(x = 35\)[/tex]:
- For [tex]\( m \angle K \)[/tex]:
[tex]\[
m \angle K = 215 - 4x = 215 - 4(35) = 215 - 140 = 75^{\circ}
\][/tex]
- For [tex]\( m \angle L \)[/tex]:
[tex]\[
m \angle L = 2x + 15 = 2(35) + 15 = 70 + 15 = 85^{\circ}
\][/tex]
- For [tex]\( m \angle M \)[/tex]:
[tex]\[
m \angle M = 140 - x = 140 - 35 = 105^{\circ}
\][/tex]
7. Compare measures of all angles:
- [tex]\( m \angle J = 95^{\circ} \)[/tex]
- [tex]\( m \angle K = 75^{\circ} \)[/tex]
- [tex]\( m \angle L = 85^{\circ} \)[/tex]
- [tex]\( m \angle M = 105^{\circ} \)[/tex]
The greatest measure is [tex]\( m \angle M = 105^{\circ} \)[/tex].
Thus, the angle with the greatest measure is [tex]\(\boxed{D \text{ (angle M)}}\)[/tex].
