The Triangle Angle Sum Theorem


One of the most elementary concepts we have learned about triangles in Geometry is the angle sum theorem. The theorem states that the sum the three interior angles of a triangle is 180^{o}. We can easily see this by duplicating or cutting the corners of a triangle and meeting the angles at a particular point (see first figure). The adjacent angles will form a straight angle which is equal to 180^{o}.

trianlge angle sum

The proof of the angle sum theorem is quite easy. We just need to draw an extra line.


Triangle ABC.


The interior angle sum of triangle ABC is equal to 180˚.


Draw a line segment passing through A and parallel to BC. This is possible because given a line and a point, you can draw exactly one line through that point parallel to the first line.

Construct points D and E to the left and right of A respectively as shown in the figure below. Now BC and DE are parallel, so AB and AC can be considered as transversals.

triangle angle sum proof


\angle BAD \cong \angle ABC since they are alternate interior angles.

Also, \angle EAC \cong \angle BCA since they are alternate interior angles.

\angle BAD, \angle BAC, and \angle EAC form a straight angle therefore their sum is 180^{o} .

But these three angles are also the interior angles of triangle ABC (Can you see why?)

Therefore, the interior angle sum of triangle ABC is 180^{o}.


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