DF−→− bisects ∠EDG
angle ef is equal to n+9
angle fg is equal to 4n-6
To find the value of n and the measures of angles EF and FG, we can use the Angle Bisector Theorem. According to the theorem, the ratio of the lengths of the two segments created by an angle bisector is equal to the ratio of the measures of the two angles formed by the angle bisector.
In this case, we have:
EF/FG = angle EDF / angle FDG
Since angle EDF and angle FDG together form angle EDG, which is bisected by DF, we can write:
EF/FG = angle EDG / angle FDG
Given that angle EDG is bisected by DF, we know that angle EDG is equal to angle FDG. Therefore, we can substitute angle FDG with angle EDG in the equation:
EF/FG = angle EDG / angle EDG
Simplifying the equation:
EF/FG = 1
This means that EF is equal in length to FG. Therefore, EF = FG.
Now, let's use the given angle measurements to solve for n and find the values of angles EF and FG.
Angle EF = n + 9
Angle FG = 4n - 6
Since EF = FG, we can set the two expressions equal to each other:
n + 9 = 4n - 6
Bringing the n terms to one side:
3n = 15
Dividing by 3:
n = 5
Now that we have found the value of n, we can substitute it back into the angle measurements:
Angle EF = 5 + 9 = 14
Angle FG = 4(5) - 6 = 14
Therefore, angles EF and FG both measure 14 degrees.