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

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle3$ are vertical angles, and $\angle2$ and $\angle4$ are vertical angles. Also, $\angle1+\angle2 = 180^{\circ}$ (linear - pair of angles). Since $\angle1=(3x - 1)^{\circ}$ and $\angle3=(3x - 1)^{\circ}$, $\angle2=(2x + 9)^{\circ}$ and $\angle4=(2x + 9)^{\circ}$, and $\angle1+\angle2=180^{\circ}$, we have the equation $(3x - 1)+(2x + 9)=180$.

Step2: Simplify the equation

Combine like - terms: $3x+2x-1 + 9=180$, which simplifies to $5x+8 = 180$.

Step3: Solve for x

Subtract 8 from both sides: $5x=180 - 8=172$. Then $x=\frac{172}{5}=34.4$.

Step4: Find the measure of $\angle1$ and $\angle3$

Substitute $x = 34.4$ into the expression for $\angle1$ (and $\angle3$ since they are equal). $\angle1=\angle3=3x-1=3\times34.4-1=103.2 - 1=102.2$.

Step5: Find the measure of $\angle2$ and $\angle4$

Substitute $x = 34.4$ into the expression for $\angle2$ (and $\angle4$ since they are equal). $\angle2=\angle4=2x + 9=2\times34.4+9=68.8+9=77.8$.

Answer:

$\angle1$ and $\angle3$ measure $102.2$ degrees. $\angle2$ and $\angle4$ measure $77.8$ degrees.