ExpandStr\( \mathrm { \displaystyle\frac{180 \cdot \left(n - 2\right)}{n} = 165 } \)ProcessLogApply the cross-multiplication rule to eliminate the denominators: \n \( \mathrm { \displaystyle\frac{a}{b} = \displaystyle\frac{c}{d} \implies a \cdot d = b \cdot c } \) \n \( \mathrm { \displaystyle\frac{180 \cdot \left(n - 2\right)}{n} = 165 \implies180 \cdot \left(n - 2\right) = 165 \cdot n } \)Str\( \mathrm { 180 \cdot \left(n - 2\right) = 165 \cdot n } \)ProcessLogUse the distributive property of multiplication: \n \( \mathrm { a \cdot ( b + c ) = a \cdot b + a \cdot c } \) \n \( \mathrm { 180 \cdot \left(n - 2\right) = -360 + 180 \cdot n } \)Str\( \mathrm { -360 + 180 \cdot n = 165 \cdot n } \)ProcessLogSet right side of the relation to zeroStr\( \mathrm { -360 + 180 \cdot n - 165 \cdot n = 0 } \)ProcessLog\( \mathrm { 180 \cdot n - 165 \cdot n = 15 \cdot n } \)Str\( \mathrm { -360 + 15 \cdot n = 0 } \)ProcessLogAdd \( \mathrm { 360 } \) to both sidesStr\( \mathrm { -360 + 15 \cdot n + 360 = 0 + 360 } \)ProcessLog\( \mathrm { -360 + 360 = 0 } \)Str\( \mathrm { 0 + 15 \cdot n = 0 + 360 } \)ProcessLogMultiply both sides with \( \mathrm { \displaystyle\frac{1}{15} } \)Str\( \mathrm { \displaystyle\frac{15}{15} \cdot n = 360 \cdot \displaystyle\frac{1}{15} } \)StrProcessLog\( \mathrm { \displaystyle\frac{15}{15} \cdot n = 360 \cdot \displaystyle\frac{1}{15} } \) \( \mathrm { \Longrightarrow } \) \( \mathrm { n = 24 } \)Str\( \mathrm { n = 24 } \)\( \mathrm { \displaystyle\frac{180 \cdot \left(n - 2\right)}{n} = 165 } \)\( \mathrm { n = 24 } \)Isolate \( \mathrm { n } \)

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Solution: A regular polygon has interior angle 165^\circ. Find the number of sides.

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