Question

what are the numerical measures of each angle in the diagram?
∠1 and ∠3 measure \boxed{} degrees.
∠2 and ∠4 measure \boxed{} degrees.

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So \(2x + 9=3x - 1\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides: \(9=x - 1\). Then add \(1\) to both sides: \(x = 10\).

Step3: Find \(\angle1\) and \(\angle3\)

Substitute \(x = 10\) into \(\angle1=(3x - 1)^{\circ}\), \(\angle1=(3\times10 - 1)^{\circ}=29^{\circ}\). Since \(\angle3=(2x + 9)^{\circ}\), \(\angle3=(2\times10 + 9)^{\circ}=29^{\circ}\).

Step4: Find \(\angle2\) and \(\angle4\)

\(\angle2\) and \(\angle1\) are supplementary (linear - pair). So \(\angle2 = 180^{\circ}-\angle1\). \(\angle2=180 - 29=151^{\circ}\). \(\angle4=\angle2 = 151^{\circ}\) (vertical angles).

Answer:

\(\angle1\) and \(\angle3\) measure \(29\) degrees. \(\angle2\) and \(\angle4\) measure \(151\) degrees.