Prove that the sum of the interior angles in an N -sided...

Key Concepts

Convex Polygon

A convex polygon is a simple polygon in which all interior angles are less than 180 degrees and every line segment between two vertices remains inside or on the boundary of the polygon. This property is crucial because it ensures that the polygon does not have any indentations, allowing for straightforward geometric constructions such as drawing diagonals that all lie completely within the polygon.

Triangulation

Triangulation is the process of dividing a polygon into a set of non-overlapping triangles by drawing non-intersecting diagonals from one vertex to all other non-adjacent vertices. This method is significant because it simplifies the problem of finding the sum of interior angles by reducing it to summing the angles of triangles, which is a well-understood concept.

Sum of Angles in a Triangle

In Euclidean geometry, the sum of the interior angles of a triangle is always 180 degrees. This fundamental theorem is pivotal when combined with triangulation because, after decomposing a convex polygon into triangles, the total sum of the interior angles of the polygon can be determined by multiplying the number of triangles formed by 180 degrees.

Transcript

00:01 So the task of this question is to prove that for any polyion with a number of sides greater than two, the sum of its interior angle measures is n -minus 2 times 180, where n is the number of sides.

00:18 So we will do this by proof by induction.

00:22 So there are essentially three steps in doing this, which is proving the base case, hypothesizing an inductive hypothesis, and then proving the.

00:32 The inductive case.

00:34 So proving the base case in this case is proving that the statement is true for the polygon with the least number of sides and this is a triangle.

00:43 So by definition, a triangle has a sum or has interior measures that sum to a measure of 180 degrees.

00:55 And we can prove that this is true.

00:57 So n is equal to 3 in this case.

00:59 So the sum of angle measures for a triangle, will be 3 minus 2 times 180, which is 1 times 180, which is equal to 180.

01:11 So we've proven the base case.

01:14 Now, to form the inductive hypothesis, we need to, for any integer number of sides, k, let's assume, or let's hypothesize that the polygon, or any polygon with a side number, k has interior angles that sum to k minus 2 times 180.

01:46 So when n is equal to k, the sum of interior angles will be equal to k minus 2 times 180.

02:05 Now for the last step, we need to prove the inductive case.

02:09 So what this step does is it proves that for any k, the statement is true for k plus 1, which means that for any integer that makes the statement true, the integer following also makes it true, and the integer following that makes it true, and so on and so forth until infinity.

02:33 So let's take a look at the polygon, which has a side length that is 3 plus 1, so the second smallest number of side length.

02:50 So we've, increase the number of side lengths by one from a triangle.

02:55 Now, if you look here, this polygon can be divided into two triangles.

03:00 And by definition, each triangle has interior angles, which sum to 180 each.

03:11 And since this polygon, this quadrilateral, can be made of two triangles, the sum of interior angles will be 180 times 2, which is 370.

03:26 Now let's look at a pentagon...

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