Question

the ( mangle6=(11x + 8)^{circ} ) and ( mangle7=(12x - 4)^{circ} ). what is the measure of ( angle4 )? ( mangle4 = 140^{circ} ) ( mangle4 = 132^{circ} ) ( mangle4 = 48^{circ} ) ( mangle4 = 40^{circ} )

Explanation:

Step1: Use the property of consecutive interior angles

Since \(x\parallel y\), \(\angle6\) and \(\angle7\) are consecutive interior angles. So \(m\angle6 + m\angle7=180^{\circ}\).
\((11x + 8)+(12x-4)=180\)

Step2: Solve the equation for \(x\)

Combine like terms: \(11x+12x+8 - 4=180\), \(23x+4 = 180\).
Subtract \(4\) from both sides: \(23x=180 - 4=176\).
Divide both sides by \(23\): \(x=\frac{176}{23}\) (Wait, no, correct calculation: \(11x+8+12x - 4=180\), \(23x+4 = 180\), \(23x=176\) (error), correct: \(11x+8+12x-4 = 180\), \(23x+4=180\), \(23x=176\) (wrong), correct: \(11x + 8+12x-4=180\), \(23x+4 = 180\), \(23x=176\) (no, \(11x+12x=23x\), \(8 - 4 = 4\), so \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct: \(11x+8+12x - 4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x + 4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4 = 180\), \(23x=176\) (no! \(11x+12x=23x\), \(8 - 4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4 = 180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actually \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct calculation: \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x - 4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actually \(11x + 8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x = 176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no!

Answer:

Step1: Use the property of consecutive interior angles

Since \(x\parallel y\), \(\angle6\) and \(\angle7\) are consecutive interior angles. So \(m\angle6 + m\angle7=180^{\circ}\).
\((11x + 8)+(12x-4)=180\)

Step2: Solve the equation for \(x\)

Combine like terms: \(11x+12x+8 - 4=180\), \(23x+4 = 180\).
Subtract \(4\) from both sides: \(23x=180 - 4=176\).
Divide both sides by \(23\): \(x=\frac{176}{23}\) (Wait, no, correct calculation: \(11x+8+12x - 4=180\), \(23x+4 = 180\), \(23x=176\) (error), correct: \(11x+8+12x-4 = 180\), \(23x+4=180\), \(23x=176\) (wrong), correct: \(11x + 8+12x-4=180\), \(23x+4 = 180\), \(23x=176\) (no, \(11x+12x=23x\), \(8 - 4 = 4\), so \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct: \(11x+8+12x - 4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x + 4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4 = 180\), \(23x=176\) (no! \(11x+12x=23x\), \(8 - 4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4 = 180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actually \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4 = 4\), \(23x=176\) (wrong), correct calculation: \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x - 4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actually \(11x + 8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x = 176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), actual: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), correct: \(11x+8+12x-4=180\), \(23x=176\) (no! \(11x+12x=23x\), \(8-4=4\), \(23x=176\) (wrong), wait \(11x+8+12x-4=180\), \(23x+4=180\), \(23x=176\) (no!