Question

classifying quadrilaterals
applying the quadrilateral angle sum theorem
what is the greatest angle measure in the diagram?

Explanation:

Step1: Apply the Quadrilateral Angle Sum Theorem

The sum of the interior angles of a quadrilateral is \(360^{\circ}\). So, \((15x - 10)+(15x + 14)+(11x + 10)+(8x - 4)=360\).
Combine like - terms: \((15x+15x + 11x+8x)+(-10 + 14+10 - 4)=360\), which simplifies to \(49x + 10=360\).

Step2: Solve for \(x\)

Subtract \(10\) from both sides: \(49x=360 - 10=350\).
Divide both sides by \(49\): \(x=\frac{350}{49} = 7.142857\approx7\).

Step3: Calculate each angle

• For \(\angle E=(15x - 10)^{\circ}\), substitute \(x = 7\): \(15\times7-10=105 - 10 = 95^{\circ}\).
• For \(\angle F=(15x + 14)^{\circ}\), substitute \(x = 7\): \(15\times7+14=105+14 = 119^{\circ}\).
• For \(\angle H=(11x + 10)^{\circ}\), substitute \(x = 7\): \(11\times7+10=77+10 = 87^{\circ}\).
• For \(\angle J=(8x - 4)^{\circ}\), substitute \(x = 7\): \(8\times7-4=56 - 4=52^{\circ}\).

Answer:

\(119^{\circ}\)