## 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. This theorem is a fundamental concept in geometry and is used to determine the measurements of angles in triangles.

To understand why this theorem holds true, let’s consider an example. Take any triangle and label its three angles as A, B, and C.

We can start by drawing a straight line through angle C which forms a straight line with segment AB. This creates two new angles, D and E.

Now, we have two line segments, AB and CD, intersected by a transversal, forming five angles: A, B, C, D, and E. We know that the sum of the angles on a straight line is 180 degrees, so angles C, D, and E collectively add up to 180 degrees.

Next, let’s focus on triangle ACD. Angle A and angle D are adjacent interior angles, meaning they share a common vertex and a common side. According to the interior angle sum of a triangle, the sum of angle ACD (angle A) and angle CAD (angle D) is equal to angle C (180 degrees). In other words, A + D = C.

Now, let’s look at triangle BCD. Angle B and angle D are also adjacent interior angles, so the sum of angle BCD (angle B) and angle CBD (angle D) is equal to angle C (180 degrees). In other words, B + D = C.

Combining the two statements, we have A + D = B + D, which means A = B. This tells us that the two remaining angles in the triangle are congruent.

Since we know that A + B + C = 180 degrees, we can replace A with B, giving us B + B + C = 180 degrees. Simplifying, we find 2B + C = 180 degrees.

Finally, we can use algebra to solve for the value of B by subtracting C from both sides of the equation: 2B = 180 – C. Dividing both sides by 2, we get B = (180 – C)/2. This formula allows us to find the measurement of angle B if we know the measurement of angle C.

In summary, the Triangle Sum Theorem states that the sum of the interior angles of a triangle is always equal to 180 degrees. This theorem is proven by using the fact that angles on a straight line add up to 180 degrees and by examining adjacent interior angles within a triangle.

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