Understanding the Triangle Sum Theorem: Interior Angle Sum of a Triangle is Always 180 Degrees

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always equal to 180 degrees

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always equal to 180 degrees.

To understand this theorem, let’s consider a triangle with vertices A, B, and C. The angles inside this triangle are denoted as ∠A, ∠B, and ∠C respectively.

We can start by drawing a line segment from vertex A to the point D on side BC, creating two smaller triangles: triangle ABC and triangle ABD.

Now, let’s examine the angles of these two triangles:

In triangle ABC, we have angles ∠A, ∠B, and ∠C.

In triangle ABD, we have angles ∠A, ∠B, and ∠D.

Since both triangles ABC and ABD share angles ∠A and ∠B, we can conclude that ∠C = ∠D. This means that ∠C and ∠D are congruent angles.

Now, we can use the fact that the sum of the angles in a straight line is 180 degrees. Since ∠C and ∠D form a straight line, we can say that ∠C + ∠D = 180 degrees.

Substituting ∠D with ∠C, we get ∠C + ∠C = 180 degrees, which simplifies to 2∠C = 180 degrees. Dividing both sides by 2, we find that ∠C = 90 degrees.

Therefore, the sum of the interior angles of triangle ABC (∠A + ∠B + ∠C) is ∠A + ∠B + 90 degrees. However, we know that the sum of the interior angles of a triangle is always equal to 180 degrees.

So, we can write the equation: ∠A + ∠B + 90 degrees = 180 degrees.

Simplifying this equation, we have ∠A + ∠B = 90 degrees.

Hence, the Triangle Sum Theorem states that the sum of the interior angles (∠A + ∠B + ∠C) of a triangle is always equal to 180 degrees.

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