Question

sum of angle measures: 540°. using tools find the value of b. then find the angle measures of the polygon. (2b - 90)° 90° (b + 45)° 2/3 b° b°

Explanation:

Step1: Set up the equation

The sum of the interior - angle measures of the polygon is given by the sum of the individual angle measures. So, \(90+(2b - 90)+(b + 45)+\frac{3}{2}b + b=540\).

Step2: Simplify the left - hand side of the equation

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

Step3: Isolate the term with \(b\)

Subtract 45 from both sides: \(\frac{11}{2}b=540 - 45\).
\(\frac{11}{2}b=495\).

Step4: Solve for \(b\)

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

Answer:

\(b = 90\)
The angle measures are:

• \(90^{\circ}\)
• \(2b-90=2\times90 - 90=90^{\circ}\)
• \(b + 45=90 + 45 = 135^{\circ}\)
• \(\frac{3}{2}b=\frac{3}{2}\times90 = 135^{\circ}\)
• \(b = 90^{\circ}\)