Question

Algebra and Geometry Question: Seven of the angles of a decagon have measures whose sum is complementary and exactly two are supplementary. Find the measures of these three angles.

Answer 1

From the present question, let's say that the missing angles are a, b, and c. We know that the sum of the other seven is equal to 1220°. By definition, the sum of the inner angles of any polygon is given by:

`S=(n-2)*180\degree`

In the present case, n = 10. Which means:

`\begin{gathered} S=(10-2)*180\degree \\ S=8*180\degree=1440\degree \\ S=1440\degree \end{gathered}`

From the information given, we are able to say that:

`\begin{gathered} 1440=1220+a+b+c \\ a+b+c=1440-1220=220 \\ a+b+c=220 \end{gathered}`

Here we found the first equation. The information about supplementary and complementary angles can be written as:

`\begin{gathered} a+b=90\degree \\ b+c=180\degree \end{gathered}`

Now we have the following system of equations:

`\begin{gathered} a+b+c=220\degree \\ a+b=90\degree \\ b+c=180\degree \end{gathered}`

If we substitute the second equation in the first one, we are able to find the value for c. Performing this calculation, we find the following:

`\begin{gathered} (a+b)+c=220\to90+c=220 \\ c=220-90=130\degree \\ c=130\degree \end{gathered}`

With this value, we are able to substitute the value of c in the third equation, and find the value of b, as follows:

`\begin{gathered} b+c=180\degree\to b+130\degree=180\degree \\ b=180\degree-130\degree=50\degree \\ b=50\degree \end{gathered}`

Now, we can substitute the value of b in the second equation of the system, to find the value of a, as follows:

`\begin{gathered} a+b=90\degree\to a+50\degree=90\degree \\ a=90\degree-50\degree=40\degree \\ a=40\degree \end{gathered}`

Now, from all the given information, we were able to find that, the three missing angles are:

`\begin{gathered} a=40\degree \\ b=50\degree \\ c=130\degree \end{gathered}`

Because the question did not name the angles, their name and order are not important, just the values.