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# geometry

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The measures of the interior angles of a convex decagon are 21x°, 154°, 118°, 106°, 15x°, 165°, 172°, 162°, 151°, and 160°. What is the measure of the smallest interior angle?

Apr 2, 2021

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SOLUTION: The sum of the angles of a convex polygon with \$n\$ sides will be \$180*(n-2)\$. For a decagon, the sum of the sides is \$180*8=1440\$. That means \$160+151+162+172+165+15x+106+118+154+21x=1440\$.

Moving constants to the right side and grouping gives \$x=7\$

That means the \$21x\$ angle is equal to \$21*7=147\$, and the \$15x\$ angle is equal to \$15*7=105\$. Comparing, you see that \$105\$ is the smllest angle, so your answer is \$\boxed{105^\circ}\$

Apr 2, 2021