Question

1. ∠3 and ∠4 are supplementary angles. find the measures of the angles when m∠3=(6x + 59)° and m∠4=(3x - 14)°.

Explanation:

Step1: Recall supplementary - angle property

Supplementary angles add up to 180°. So, \(m\angle3 + m\angle4=180^{\circ}\).

Step2: Substitute the given angle - measures

\((6x + 59)+(3x - 14)=180\).

Step3: Simplify the left - hand side

Combine like terms: \(6x+3x + 59 - 14=180\), which gives \(9x + 45=180\).

Step4: Solve for \(x\)

Subtract 45 from both sides: \(9x=180 - 45\), so \(9x = 135\). Then divide both sides by 9: \(x=\frac{135}{9}=15\).

Step5: Find \(m\angle3\)

Substitute \(x = 15\) into the expression for \(m\angle3\): \(m\angle3=(6\times15 + 59)^{\circ}=(90 + 59)^{\circ}=149^{\circ}\).

Step6: Find \(m\angle4\)

Substitute \(x = 15\) into the expression for \(m\angle4\): \(m\angle4=(3\times15 - 14)^{\circ}=(45 - 14)^{\circ}=31^{\circ}\).

Answer:

\(m\angle3 = 149^{\circ}\), \(m\angle4 = 31^{\circ}\)