DF bisects <EDG find FG. The diagram is not to scale
EDG x+7 2x+4
Since DF bisects ∠EDG, we can deduce that ∠EDF and ∠FDG are congruent. Therefore, we have:
∠EDF = ∠FDG
Since the sum of the angles in a triangle is 180 degrees, we can set up an equation:
(∠EDF) + (∠FDG) + (∠EDG) = 180
Substituting the given values, we have:
(x+7) + (2x+4) + (∠EDG) = 180
Combining like terms, we get:
3x + 11 + (∠EDG) = 180
Simplifying further:
3x + (∠EDG) = 169
Now, let's find the value of FG based on the angle relations in triangle DFG:
∠EDG and ∠DFG are vertical angles, so they are congruent. Therefore, we have:
∠EDG = ∠DFG
Let FG = y. Then, using the equation for the sum of the angles in triangle DFG, we have:
(∠EDG) + (∠DFG) + (∠DFG) = 180
Substituting the value of ∠EDG from earlier, we get:
∠DFG + ∠DFG + (∠DFG) = 180
Simplifying, we have:
3∠DFG = 180
Dividing both sides by 3, we find:
∠DFG = 60
Since DF bisects ∠EDG, we know that ∠EDF and ∠FDG are congruent. Let each of these angles be θ. Therefore:
∠EDF = ∠FDG = θ
And since ∠DFG = 60, we know that:
2θ + 60 = 180
Subtracting 60 from both sides, we have:
2θ = 120
Dividing both sides by 2, we find:
θ = 60
Now, we can use the value of θ to find the value of x:
∠EDF = x + 7 = 60
Subtracting 7 from both sides, we have:
x = 53
Finally, we can find FG using the value of x:
FG = 2x + 4 = 2(53) + 4 = 106 + 4 = 110
Therefore, FG is equal to 110 units.