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

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

Triangle Angle Sum Theorem: A More Friendly Proof

Introduction

In high school mathematics, we were taught to write proofs in two columns. There are some proofs however that are easier to understand if we use Algebra. The triangle angle sum theorem that I have posted yesterday can be proven algebraically. Its proof which I think is easier to understand is shown below.

triangle angle sum

Theorem

The sum of the interior angles of a triangle is $latex 180^{\circ}$ Continue reading