The Triangle Angle Sum Theorem | Explained and Proven

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the three angles in any triangle is always equal to 180 degrees

The Triangle Angle Sum Theorem states that the sum of the three angles in any triangle is always equal to 180 degrees. In other words, if you measure the angles of any triangle with a protractor, add up the measurements, and the total will always be 180 degrees.

To understand why this theorem is true, let’s consider an arbitrary triangle. We can draw a line, called a transversal, parallel to one of the sides of the triangle. This line creates a new angle by intersecting with one of the other sides of the triangle.

Now, there are two parallel lines (the side of the triangle and the transversal), so the alternate interior angle theorem tells us that the newly formed angle and the corresponding opposite interior angle (the angle between the other side of the triangle and the transversal) are congruent.

Additionally, the sum of the interior angles of a straight line is always 180 degrees. Since the two congruent angles formed by the transversal and the triangle’s sides are supplementary (add up to 180 degrees), each of those angles must be 90 degrees.

Now, consider the two right angles formed by the transversal and the triangle’s sides. The remaining angle within the triangle (not on the transversal) must be supplementary to the sum of those two right angles, i.e., its measure must be 180 degrees minus 90 degrees minus 90 degrees, which equals 0 degrees.

However, 0 degrees is not a valid angle measure, so we conclude that the only possible sum of the three angles in a triangle is 180 degrees.

Therefore, no matter the shape or size of a triangle, the sum of its interior angles will always equal 180 degrees. This property is known as the Triangle Angle Sum Theorem.

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