Углови многоугла

Текст задатака објашњених у видео предавању.

Пр.1) Израчунати збир унутрашњих углова у конвексном:

(а) седмоуглу       (б) десетоуглу      (в) седамнаестоуглу.

Пр.2) Збир унутрашњих углова у конвексном многоуглу износи:

(а)   ${720^ \circ }$    (б)   ${1800^ \circ }$.

Који је то мнноугао?

Пр.3) Да ли постоји конвексан многоугао чији је збир унутрашњих углова:

(а)   ${1440^ \circ }$    (б)   ${1000^ \circ }$?

Пр.4) Израчунати све унутрашњи углове конвексног шестоугла ако су мере четири угла ${100^ \circ }{,83^ \circ }{,102^ \circ }{,136^ \circ }$, а преостала два су међусобно једнака.

Пр.1) 

\[\begin{gathered}
{S_n} = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
\underline {n = 7} \hfill \\
{S_n} = ? \hfill \\
{S_7} = \left( {7 - 2} \right) \cdot {180^ \circ } \hfill \\
{S_7} = {900^ \circ } \hfill \\
\end{gathered} &{}&\begin{gathered}
\underline {n = 10} \hfill \\
{S_n} = ? \hfill \\
{S_7} = \left( {10 - 2} \right) \cdot {180^ \circ } \hfill \\
{S_7} = {1440^ \circ } \hfill \\
\end{gathered} &{}&\begin{gathered}
\underline {n = 17} \hfill \\
{S_n} = ? \hfill \\
{S_{17}} = \left( {17 - 2} \right) \cdot {180^ \circ } \hfill \\
{S_7} = {2700^ \circ } \hfill \\
\end{gathered}
\end{array} \hfill \\
\end{gathered} \]

Пр.2) 

\[\begin{gathered}
{S_n} = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
\underline {{S_n} = {{720}^ \circ }} \hfill \\
n = ? \hfill \\
{720^ \circ } = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
n - 2 = \frac{{{{720}^ \circ }}}{{{{180}^ \circ }}} \hfill \\
n - 2 = 4 \hfill \\
n = 6 \hfill \\
\end{gathered} &{}&\begin{gathered}
\underline {{S_n} = {{1800}^ \circ }} \hfill \\
n = ? \hfill \\
{1800^ \circ } = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
n - 2 = \frac{{{{1800}^ \circ }}}{{{{180}^ \circ }}} \hfill \\
n - 2 = 10 \hfill \\
n = 12 \hfill \\
\end{gathered} &{}&{}
\end{array} \hfill \\
\end{gathered} \]

Пр.3) 

\[\begin{gathered}
{S_n} = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
\underline {{S_n} = {{1440}^ \circ }} \hfill \\
n = ? \hfill \\
n \in \mathbb{N},n \geqslant 3 \hfill \\
{1440^ \circ } = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
n - 2 = \frac{{{{1440}^ \circ }}}{{{{180}^ \circ }}} \hfill \\
n - 2 = 8 \hfill \\
n = 8 + 2 \hfill \\
n = 10 \hfill \\
\hfill \\
\end{gathered} &{}&\begin{gathered}
\underline {{S_n} = {{1000}^ \circ }} \hfill \\
n = ? \hfill \\
n \in \mathbb{N},n \geqslant 3 \hfill \\
{1000^ \circ } = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
n - 2 = \frac{{{{1000}^ \circ }}}{{{{180}^ \circ }}} \hfill \\
n - 2 = \frac{{50}}{9} \hfill \\
n = \frac{{50}}{9} + 2 \hfill \\
n = 7\frac{5}{9} \hfill \\
\end{gathered} &{}&\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\end{gathered}
\end{array} \hfill \\
\end{gathered} \]

                           постоји                                     не постоји

Пр.4) 

\[\begin{gathered}
{S_n} = \left( {n - 2} \right) \cdot {180^ \circ } \hfill \\
{S_6} = \left( {6 - 2} \right) \cdot {180^ \circ } \hfill \\
{S_6} = 4 \cdot {180^ \circ } \hfill \\
{S_6} = {720^ \circ } \hfill \\
\end{gathered} \]

${100^ \circ } + {83^ \circ } + {102^ \circ } + {136^ \circ } + x + x = {720^ \circ } $
${421^ \circ } + 2x = {720^ \circ } $
$2x = {720^ \circ } - {421^ \circ } $
$2x = {299^ \circ } $
$x = {149^ \circ }30' $