how does the exterior angle theorem makes sense based on the triangle angle-sum theorem?
i know what both of the theorems are i just dont understand how they are 'connected'?? thanks!
The Exterior Angle Theorem and the Triangle Angle-Sum Theorem are indeed connected. To understand their connection, let's first define each theorem.
The Triangle Angle-Sum Theorem states that the sum of the interior angles of any triangle is always 180 degrees. In other words:
Angle A + Angle B + Angle C = 180 degrees, where A, B, and C are the interior angles of the triangle.
On the other hand, the Exterior Angle Theorem deals with the relationship between an exterior angle and its corresponding interior angles of a triangle. It states that an exterior angle of a triangle is always equal to the sum of the two non-adjacent interior angles.
Now let's see how these two theorems are connected:
1. Consider a triangle with interior angles A, B, and C.
2. Extend one of the sides of the triangle to form an exterior angle, let's call it E.
3. By the Exterior Angle Theorem, the exterior angle E is equal to the sum of the two non-adjacent interior angles, which in this case are angles A and B.
4. So, we can write the equation: E = A + B.
5. Now, let's consider the interior angles of the entire triangle (including the exterior angle): A, B, C, and E.
6. According to the Triangle Angle-Sum Theorem, the sum of all these interior angles is always 180 degrees.
7. So, we can write the equation: A + B + C + E = 180 degrees.
8. Now, substitute E = A + B (from step 4) into the equation: A + B + C + (A + B) = 180 degrees.
9. Simplify the equation: 2A + 2B + C = 180 degrees.
10. Divide both sides of the equation by 2: A + B + (C/2) = 90 degrees.
11. Rearrange the equation: A + B = 90 - (C/2).
12. This equation shows that the sum of the two non-adjacent interior angles (A and B) is equal to the difference between 90 degrees and half of the third (adjacent) interior angle (C/2).
So, by connecting the Exterior Angle Theorem and the Triangle Angle-Sum Theorem, we can see that the sum of the two non-adjacent interior angles of a triangle is equal to the difference between 90 degrees and half of the third (adjacent) interior angle.