Linking Triangle Angle Sum and Inscribed Angle Theorems

We have shown that the angle sum of a triangle is $latex 180^\circ$. We have also shown that the measure of an inscribed angle is half the measure the central angle that intercepts the same arc. In this post, we use the Inscribed Angle Theorem to show that the Triangle Angle Sum Theorem holds.

central-angle

Theorem

The angle sum of a triangle is $latex 180^\circ$.

Proof

Consider the figures above. In the first figure, the triangle is divided into three central angles. Clearly the three angles add up to a complete rotation about the center so  Continue reading

The Proof of the Polygon Angle Sum Theorem

Introduction

We have learned that the angle sum of a triangle is $latex 180^\circ$. We have also learned that  the angle sum of a quadrilateral is $latex 360^\circ$. In getting the angle sum of quadrilaterals, we divided the quadrilateral into two triangles by drawing a diagonal. In this post, we use this method to find the angle sum of the pentagon and other polygons.

Let us extend the method stated above to pentagon (5-sided polygon). Clearly, we can divide the pentagon into three non-overlapping triangles by drawing two diagonals. Since each triangle has an angle sum of $latex 180^\circ$, the angle sum of a pentagon, which is composed of three triangles, is $latex 540^\circ$.

Polygon Angle Sum

Using the method above, we can see the pattern on the table below. The sum of a polygon with $latex n$ sides is $latex 180(n-2)$ degrees. Next, we summarize the polygon angle sum theorem and prove it. Continue reading

Proof of Angle Sum of Quadrilaterals

Introduction

We have learned that the angle sum of a triangle is $latex 180^\circ$. What about the quadrilaterals? The square and the rectangles have four right angles, so clearly, the sum of their interior angles is $latex 360^{\circ}$. But what about other quadrilaterals?

Quadrilaterals

image via Wikipedia

Before proceeding with the proof, to those who want to explore first, draw different types of quadrilaterals and use the protractor to measure their interior angles. What do you think is the angle sum of the quadrilaterals? Are the sums equal? Continue reading