Question

of the variable. then find the angle measures of the polygon. answer. 30. sum of angle measures: 540°

Explanation:

Step1: Set up equation

The sum of interior - angles of a polygon is given. We add up all the angle expressions and set it equal to the sum. So, $b+\frac{3}{2}b+(b + 45)+(2b-90)+90 = 540$.

Step2: Combine like - terms

Combine the $b$ terms: $(1+\frac{3}{2}+1 + 2)b+(45-90 + 90)=540$. First, $1+\frac{3}{2}+1 + 2=\frac{2 + 3+2 + 4}{2}=\frac{11}{2}$. The equation becomes $\frac{11}{2}b + 45=540$.

Step3: Isolate the variable term

Subtract 45 from both sides: $\frac{11}{2}b=540 - 45$. So, $\frac{11}{2}b=495$.

Step4: Solve for $b$

Multiply both sides by $\frac{2}{11}$: $b = 495\times\frac{2}{11}=90$.

Step5: Find angle measures

• First angle: $b = 90^{\circ}$.
• Second angle: $\frac{3}{2}b=\frac{3}{2}\times90 = 135^{\circ}$.
• Third angle: $b + 45=90 + 45=135^{\circ}$.
• Fourth angle: $2b-90=2\times90-90 = 90^{\circ}$.
• Fifth angle: $90^{\circ}$.

Answer:

$b = 90^{\circ}$, and the angle measures are $90^{\circ},135^{\circ},135^{\circ},90^{\circ},90^{\circ}$