## Why is the angle sum of the interior angles of a triangle 180 degrees?

We were taught that the sum of the measures of interior angles of a triangle is 180 degrees. But how come? Is it true no matter what the shape or size of the triangle?

Figure 1 – Triangles with interior angles a, b and c

Recall that an angle is the amount of rotation of a ray.  In Figure 2, the ray was rotated from A to C, and the amount of rotation is 60 degrees. We can say that the measure of angle ABC is 60 degrees.

Figure 2 – Ray containing A is rotated to to C making a 60-degree angle rotation

The rotation from A to D forms a straight line and measures 180 degrees.  Therefore, straight angle ABD measures 180 degrees.  It follows that a 180-degree rotation is a half-circle. Therefore, a complete rotation is 360 degrees.

We can verify if our question about the sum of the interior angles of a triangle by drawing a triangle on a paper, cutting the corners, meeting the corners (vertices) at one point  such that the sides coincide with no gaps and overlaps (see Figure 3). Notice that no matter what the size or shape of a triangle, as long as the previous conditions are met, the two of its sides will be collinear as shown in the Figure below.

Figure 4 – The corners of a triangle put together

However, this is not the proof. To discuss the proof, we are going to use Euclid’s fifth postulate. Euclid’s fifth postulate tells us that if a parallel line is cut by a transversal, their corresponding angles are congruent. In the diagram below, lines p and q are parallel lines and angles shown with the same letters are corresponding angles, and hence congruent.

Figure 5 – Parallel lines p and q cut by a transversal.

Now how do we connect Parallel Postulate to the angle sum of a triangle. Consider a triangle with interior angles a, b, and c. Using the fifth postulate, we use the side containing angles a and c as base of the triangle and extend it to both sides. Next, we draw a line to the base and passing thorugh the vertex not on the base.

Figure 6 – Triangle with three points on two parallel lines

Now, the sum of angles a, b and c is 180 degrees (why?). But a, b and c are also the interior angles of a polygon. Therefore, the sum of the interior angles of a triangle is 180 degrees.

You may also want to read Sum of the Interior Angles of a Polygon.

## Sum of the interior angles of a polygon

We were taught that if we let $m$ be the angle sum (the total measure of the interior angles) and $n$ be the number of vertices (corners)  of a polygon, then $m = 180(n-2)$.  For example, a quadrilateral has $4$ vertices, so its angle sum is $180(4-2) = 360$ degrees.  Similarly, the angle sum of a hexagon (a polygon with $6$ sides) is $180(6-2) = 720$ degrees.

But where did this formula come from?  Does this formula work for all polygons?  (Note that in this discussion, when we say polygon, we only refer to convex polygons).

Before we answer these questions, let us first have a brief review of some elementary concepts.

Polygons and Interior Angles

A polygon is a closed figure with finite number of sides. In the figures below, $ABCDE$ is a polygon with $5$ sides and ($5$ vertices).  It is clear that the number of sides of a polygon is always equal to the number of its vertices.

A polygon has interior angles.  In the first figure below, angle $B$ measuring $91$ degrees is an interior angle of polygon $ABCDE$. The angle sum $m$ of $ABCDE$ (not drawn to scale) is given by the equation

$m = 126 + 91 + 113 + 102 + 108 = 540$ degrees.

In the second figure, if we let $a_1, a_2$ and $a_3$ be the measure of the interior angles of triangle $ABE$, then the angle sum m of triangle $ABE$ is given by the equation $m = a_1 + a_2 + a_3$.

Angle Sum

To generalize our calculation of  angle sum, we use the fact that the angle sum of a triangle is $180$ degrees. Notice that any polygon maybe divided into triangles by drawing diagonals from one vertex to all of the non-adjacent vertices.  In the second figure above, the pentagon was divided into three triangles by drawing diagonals from vertex $E$ to the non-adjacent vertices $B$ and $C$ forming $BE$ and $CE$.  Now let $a_k, b_k$ and $c_k$, where $k = 1, 2, 3$ be measures of the interior angles of the three triangles as shown on the second figure.

Calculating the angle sum of pentagon $ABCDE$ we have

Notice that the angle measures in the first line of our equation is just a rearrangement of the measures of the interior angles of the three triangles. Hence, the angle sum of the pentagon is equal to the angle sum of the three triangles. Therefore, we can conclude that the sum of the interior angles of a polygon is equal to the angle sum of the number of triangles that can be formed by dividing it using the method described above. Using this conclusion, we will now relate the number of sides of a polygon, the number of triangles that can be formed by drawing diagonals and the polygon’s angle sum.

From the table above, we observe that the number of triangles formed is $2$ less than the number of sides of the polygon.  This is true, because $n - 2$ triangles can be formed by drawing diagonals from one of the vertices to $n - 3$ non-adjacent vertices. Therefore, there the angle sum $m$ of a polygon with $n$  sides is given by the formula

$m = 180(n - 2)$

A More Formal Proof

Theorem: The sum of the interior angles of a polygon with $n$ sides is $180(n-2)$ degrees.

Proof:

Assume a polygon has $n$ sides. Choose an arbitrary vertex, say vertex $V$.  Then there are $n - 3$ non-adjacent vertices to vertex $V$.  If diagonals are drawn from vertex $V$ to all non-adjacent vertices, then $n - 2$ triangles will be formed.  The sum the interior angles of $n -2$ triangles is $180(n - 2)$. Since the angle sum of the polygon with $n$ sides is equal to the sum the interior angles of $n - 2$ triangles, the angle sum of a polygon with $n$ sides is $180(n-2)$. $\blacksquare$

Exercises:

1.)    Find the number the angle sum of a dodecagon ($12$-sided polygon).

2.)    The angle sum of a polygon is $3240$ degrees. What is the number of its sides?

3.)    The measure of one of the angles of a regular polygon is $150$. Find its number of sides.

4.)    From this, prove that the sum of the interior angles of a polygon is $360$ degrees.

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