The Importance Of Corollaries In Mathematics: The Consequences Of Proven Propositions

corollaries to the triangle sum theorem

The acute angles of a right triangle are complementary. The measure of each angle of an equiangular triangle is 60 degrees.

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always equal to 180 degrees. There are several corollaries to this theorem that can be derived from it. Some of the most important corollaries are:

1. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. In other words, if you extend one side of a triangle, then the angle formed outside the triangle is equal to the sum of the interior angles that are not adjacent to it.

2. Corollary 1: If two angles of a triangle are equal, then the third angle is also equal to them. This corollary follows from the fact that the sum of the interior angles of a triangle is always equal to 180 degrees.

3. Corollary 2: If one angle of a triangle is acute, then the other two angles are also acute. This corollary also follows from the fact that the sum of the interior angles is 180 degrees. If one angle is acute (less than 90 degrees), then the other two angles must also be less than 90 degrees to add up to 180 degrees.

4. Corollary 3: If one angle of a triangle is obtuse, then the other two angles are acute. Again, this follows from the fact that the sum of the interior angles is always 180 degrees. If one angle is obtuse (greater than 90 degrees), then the other two angles must be less than 90 degrees to add up to 180 degrees.

5. Corollary 4: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. This corollary is known as the Triangle Inequality Theorem. It can be proven using the fact that the shortest distance between two points is a straight line. If the sum of two sides is less than the third side, then the three points cannot form a triangle, and the shortest distance between two of the points would not be a straight line.

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