By definition, an angle bisector divides an angle into two equal parts. Thus, ∠AEB ≅ ∠BEC by definiton.
Because EA is perpendicular to ED, it makes a right angle at E. Thus the measure of ∠AED is 90°.
Using these two pieces of information - the definition of an angle bisector and the definition of perpendicular, we can find the measure of the pieces inside by angle addition.
Now we take those two definitions and our given angles to find x. Thus,
m∠AEB + m∠BEC + m ∠CED = m∠AED by angle addition
m∠AEB + m∠BEC + m ∠CED = 90 by definition of perpendicular
m∠AEB + m∠AEB + m ∠CED = 90 by defintion of angle bisector for EA.
(4x + 1) + (4x +1) + m ∠CED = 90 because m∠AEB =4x + 1 is given
4x +1 + 4x +1 + 3x = 90 because m∠CED =3x is given
11x + 2 = 90 by collecting like terms
11x = 88 by subtracting 2 on both sides
x = 8 by dividing both sides by 11.
Now we can substitute to find the angles.
Since m∠AEB is 4x + 1, then m∠AEB = 4(8) + 1 = 32 + 1 = 33°. Since m∠CED is 3x, then m∠CED is 3(8) = 24 degrees. The remaining angle, ∠BEC, was equal to ∠AEB by definition of angle bisector. It too is 33 degrees.
Therefore, x = 8, m∠AEB = m∠BEC = 33° and m∠CED = 24°
