Part 1: The two equations that can be used to determine the measure of these two adjacent angles, which add up to 180 degrees, are:
- (2x + 3) + (x - 6) = 180 (This correctly represents the sum of the two angles.)
- (2x + 3) - (x - 6) = 180 (This does not correctly represent the angles but can be manipulated correctly.)
So, the correct choice for determining the adjacent angles is:
(2x + 3) + (x - 6) = 180.
Part 2: Let's solve the first equation:
(2x + 3) + (x - 6) = 180
Combine like terms:
2x + 3 + x - 6 = 180 3x - 3 = 180
Now add 3 to both sides:
3x = 180 + 3 3x = 183
Now divide by 3:
x = 61.
Now that we have the value of x, we can find the measures of the angles:
-
Angle \( 2x + 3 \):
- \( 2(61) + 3 = 122 + 3 = 125 \) degrees.
-
Angle \( x - 6 \):
- \( 61 - 6 = 55 \) degrees.
Final answers:
- x = 61
- Angle \( 2x + 3 \) is 125 degrees.
- Angle \( x - 6 \) is 55 degrees.