We are asked to show that in an octagon the sum of the angle measures is $(8 - 2) 180 .$ Let's draw an octagon with all diagonals from one of its vertices. Let's label all angles on the diagram.

The sum of the measures of all labeled angles gives the sum of all interior angle measures of the octagon. This is illustrated in the following table. Adding the angle measures in the columns first gives the angle measures at the vertices. Adding these column sums gives the sum of all interior angle measures.

Notice that the table also groups the angles in the rows as the angles of triangles. According to the Triangle Angle Sum Theorem, these row sums are all

180 .

There are six rows in the table, so we can write the sum of these row sums as a product.

6 \times 180 = (8 - 2) 180

Since the sum of the row sums is the same as the sum of the column sums, this proves that the sum of the interior angle measures in an octagon is

(8 - 2) 180 .

We can summarize the argument above in a paragraph proof.

Completed proof

Claim: The sum of the angle measures in an octagon is $(8 - 2) 180 .$

Proof: The five diagonals from one vertex of an octagon cut it into six triangles. These diagonals also cut the interior angles of the octagon. These angle parts are angles of the triangles. There are six triangles and in all of these the sum of the angle measures is $180 .$ Adding these together, we get that the sum of the interior angle measures of an octagon is $6 \times 180 = (8 - 2) 180 .$

Exercise

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