Use the Triangle Angle Sum Theorem to find the measure of the angle exterior to the smallest angle of a triangle with angle measures 2 x + 6, 4x - 1, and 6x + 7
The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees.
Let's set up an equation using this theorem:
(2x + 6) + (4x - 1) + (6x + 7) = 180
Combine like terms:
12x + 12 = 180
Subtract 12 from both sides:
12x = 168
Divide both sides by 12:
x = 14
Now we can substitute this value back into the given angle measures to find the smallest angle:
2x + 6 = 2(14) + 6 = 34 degrees
We want to find the measure of the exterior angle that is opposite the smallest angle. Exterior angles are formed when a side of the triangle is extended. The measure of an exterior angle is the sum of the two opposite interior angles. In this case, the exterior angle is opposite the smallest angle, so we will add the other two angles:
(4x - 1) + (6x + 7) = 4(14) - 1 + 6(14) + 7 = 56 - 1 + 84 + 7 = 146 degrees
Therefore, the measure of the angle exterior to the smallest angle is 146 degrees.