Geometry Practice Test 3

Internal angles of the first regular polygon = 2x
Internal angles of the second regular polygon = 3x
2x + E = 180
E = 180 - 2x, where E is the external angle of regular polygon
Similarly, E = 180 - 3x
Number of sides of the 1st polygon = $\frac{360}{180-2\mathit{x}}$
Number of sides of the other polygon = $\frac{360}{180-3\mathit{x}}$
Ratio of the number of sides = 1 : 2
$\frac{\frac{360}{180-2\mathit{x}}}{\frac{360}{180-3\mathit{x}}}$ = $\frac{1}{2}$
$\frac{360}{180-2\mathit{x}}\times \frac{180-2\mathit{x}}{360}$ = $\frac{1}{2}$
360 − 6x = 180 − 2x
x = $\frac{180}{4}$
x = 45
Now number of sides of the first polygon
= $\frac{360}{180-90}$ = $\frac{360}{90}$ = 4
Other sides = $\frac{360}{180-145}$ = $\frac{360]}{45}$ = 8
Number of sides = 4, 8