Question

applying the quadrilateral angle sum theorem
what is the greatest angle measure in the diagram?
95°
105°
119°
180°

Explanation:

Step1: Recall the quadrilateral angle - sum theorem

The sum of the interior angles of a quadrilateral is \(360^{\circ}\). So, \((15x - 10)+(15x + 14)+(11x + 10)+(9x - 4)=360\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((15x+15x + 11x+9x)+(-10 + 14+10 - 4)=360\).
\(50x + 10=360\).

Step3: Solve for \(x\)

Subtract \(10\) from both sides: \(50x=360 - 10=350\).
Divide both sides by \(50\): \(x=\frac{350}{50}=7\).

Step4: Find the measure of each angle

• For \(\angle E=(15x - 10)^{\circ}\), substitute \(x = 7\): \(\angle E=(15\times7-10)^{\circ}=(105 - 10)^{\circ}=95^{\circ}\).
• For \(\angle F=(15x + 14)^{\circ}\), substitute \(x = 7\): \(\angle F=(15\times7+14)^{\circ}=(105 + 14)^{\circ}=119^{\circ}\).
• For \(\angle H=(11x + 10)^{\circ}\), substitute \(x = 7\): \(\angle H=(11\times7+10)^{\circ}=(77 + 10)^{\circ}=87^{\circ}\).
• For \(\angle G=(9x - 4)^{\circ}\), substitute \(x = 7\): \(\angle G=(9\times7-4)^{\circ}=(63 - 4)^{\circ}=59^{\circ}\).

Answer:

\(119^{\circ}\) (corresponding to the option \(119^{\circ}\))