To solve this problem, we need to determine the measure of each angle using the given relationships and find which angle has the greatest measure.
1. 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]
2. Properties of a Quadrilateral Inscribed in a Circle:
Quadrilateral JKLM is inscribed in a circle, meaning the sum of the measures of each pair of opposite angles equals [tex]\(180^\circ\)[/tex].
- [tex]\( m_{\angle J} + m_{\angle L} = 180^\circ \)[/tex]
- [tex]\( m_{\angle K} + m_{\angle M} = 180^\circ \)[/tex]
3. Setting Up Equations:
For [tex]\( m_{\angle J} + m_{\angle L} = 180^\circ \)[/tex]:
[tex]\[
95 + (2x + 15) = 180
\][/tex]
Simplifying:
[tex]\[
95 + 2x + 15 = 180
\][/tex]
[tex]\[
2x + 110 = 180
\][/tex]
[tex]\[
2x = 70
\][/tex]
[tex]\[
x = 35
\][/tex]
For [tex]\( m_{\angle K} + m_{\angle M} = 180^\circ \)[/tex]:
[tex]\[
(215 - 4x) + (140 - x) = 180
\][/tex]
Simplifying:
[tex]\[
215 - 4x + 140 - x = 180
\][/tex]
[tex]\[
355 - 5x = 180
\][/tex]
[tex]\[
-5x = -175
\][/tex]
[tex]\[
x = 35
\][/tex]
4. Calculating Angle Measures:
Using [tex]\( x = 35 \)[/tex]:
- [tex]\( m_{\angle J} = 95^\circ \)[/tex] (given directly)
- [tex]\( m_{\angle K} = 215 - 4(35) = 215 - 140 = 75^\circ \)[/tex]
- [tex]\( m_{\angle L} = 2(35) + 15 = 70 + 15 = 85^\circ \)[/tex]
- [tex]\( m_{\angle M} = 140 - 35 = 105^\circ \)[/tex]
5. Comparing Angle Measures:
- [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]
Among these values, the greatest angle measure is [tex]\( 105^\circ \)[/tex].
6. Conclusion:
The angle with the greatest measure is [tex]\( m_{\angle M} = 105^\circ \)[/tex].
Therefore, the correct answer is:
A. angle [tex]\( M \)[/tex]
