Question

quadrilateral jklm is inscribed in a circle. quadrilateral jklms angles are described as follows: m∠j = 95°, m∠k=(215 - 4x)°, m∠l=(2x + 15)°, m∠m=(140 - x)°. which angle has the greatest measure? a. angle l b. angle m c. angle k d. angle j

Explanation:

Step1: Use the property of inscribed quadrilateral

The sum of opposite angles of a cyclic quadrilateral is 180°. But we can also use the fact that the sum of the interior angles of a quadrilateral is 360°. So, \(m\angle J+m\angle K + m\angle L+m\angle M=360^{\circ}\). Substitute the given angle - measures: \(95+(215 - 4x)+(2x + 15)+(140 - x)=360\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((95 + 215+15 + 140)+(-4x+2x - x)=360\).
\(465-3x = 360\).

Step3: Solve for \(x\)

Subtract 465 from both sides: \(-3x=360 - 465=-105\).
Divide both sides by \(-3\): \(x = 35\).

Step4: Find the measure of each angle

• \(m\angle J=95^{\circ}\).
• \(m\angle K=215-4x\), substitute \(x = 35\), then \(m\angle K=215-4\times35=215 - 140 = 75^{\circ}\).
• \(m\angle L=2x + 15\), substitute \(x = 35\), then \(m\angle L=2\times35+15=70 + 15=85^{\circ}\).
• \(m\angle M=140 - x\), substitute \(x = 35\), then \(m\angle M=140-35 = 105^{\circ}\).

Answer:

C. angle M