Question

1. what is the complement of a 40° angle?
2. what is the supplement of a 135° angle?
3. in the accompanying diagram, the adjacent angles formed by intersecting (overline{ab}) and (overline{cd}) have measures ((3x + 50)^{circ}) and ((x + 10)^{circ}). solve for (x).

Explanation:

Step1: Recall the definition of complementary angles

Complementary angles add up to 90°. Let the complement of a 40 - degree angle be \(x\). Then \(x + 40^{\circ}=90^{\circ}\).

Step2: Solve for \(x\)

Subtract 40° from both sides of the equation: \(x=90^{\circ}- 40^{\circ}=50^{\circ}\).

Step3: Recall the definition of supplementary angles

Supplementary angles add up to 180°. Let the supplement of a 135 - degree angle be \(y\). Then \(y + 135^{\circ}=180^{\circ}\).

Step4: Solve for \(y\)

Subtract 135° from both sides of the equation: \(y=180^{\circ}-135^{\circ}=45^{\circ}\).

Step5: Use the property of adjacent angles

Adjacent angles formed by intersecting lines \(\overline{AB}\) and \(\overline{CD}\): The sum of adjacent angles is 180°. So, \((3x + 50)+(x + 10)=180\).

Step6: Combine like - terms

\(3x+x+50 + 10=180\), which simplifies to \(4x+60 = 180\).

Step7: Isolate the variable term

Subtract 60 from both sides: \(4x=180 - 60=120\).

Step8: Solve for \(x\)

Divide both sides by 4: \(x=\frac{120}{4}=30\).

Answer:

1. \(50^{\circ}\)
2. \(45^{\circ}\)
3. \(x = 30\)