Linking Triangle Angle Sum and Inscribed Angle Theorems

We have shown that the angle sum of a triangle is 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 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 

measure of blue angle + measure of red angle + measure of green angle = 360^\circ.

In the second figure, the colored angles are inscribed angles. The measure of each angle is half the measure of its corresponding central angle (the angle of the same color) in the first figure (see also third figure). That is, the measure of the blue angle in the second figure is half the measure of the blue angle in the first figure. This means that the sum of measures of the inscribed angles equal to

1/2(measure of blue angle) + 1/2(measure of red angle) + 1/2(measure of green angle)

of the angles in the first figure

or

1/2(measure of blue angle + measure of red angle + measure of green angle).

of the angles in the first figure.

Now, this is equaivalent to  1/2 the sum of the measures of the central angles or \frac{1}{2} (360^\circ) = 180^\circ.

But the sum of the inscribed angles above is equal to the sum of the interior angles of triangle ABC in the second figure, therefore, the angle sum of triangle ABC is 180^\circ.

Now, since any triangle can be inscribed in a circle, the sum of the interior angles of any triangle is equal to 180^\circ.

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