The value of x is 6.7°.
Since ABCD is a rhombus, all of its sides are equal. We are given that the diagonals intersect at point E.
We are also given that AE = 23x and EC = x - 10. We need to find the value of x.
From the diagram, we can see that triangle ABE is a right triangle, with angle AEB being a right angle.
We are also given that m/AEB = (16x + 6)°.
Using the Pythagorean Theorem, we can find the length of AB:
AB^2 = AE^2 + EB^2
Since ABCD is a rhombus, we know that AB = AD. Substituting this into the equation, we get:
AD^2 = AE^2 + EB^2
We are given that AE = 23x and EC = x - 10. Substituting these values into the equation, we get:
AD^2 = (23x)^2 + (x - 10)^2
Simplifying the equation, we get:
AD^2 = 529x^2 + 40x - 100
We are also given that m/AFB = (16x + 6)°. Since ABCD is a rhombus, we know that AFB = 90°. Substituting this into the equation, we get:
m/AEB = 180° - m/AFB
m/AEB = 180° - (16x + 6)°
m/AEB = 174° - 16x
Since triangle AEB is a right triangle, we know that the sum of the angles in the triangle is 180°. Therefore, we can write the following equation:
m/AEB + m/ABE + m/BAE = 180°
Substituting the values we know, we get:
174° - 16x + m/ABE + m/BAE = 180°
Combining like terms, we get:
m/ABE + m/BAE = 6x + 6°
Since diagonals of a rhombus bisect opposite angles, we know that m/ABE = m/BAE. Therefore, we can rewrite the equation as:
2 * m/ABE = 6x + 6°
m/ABE = 3x + 3°
We also know that m/AEB = 90° - m/ABE. Substituting the value we found for m/ABE, we get:
m/AEB = 90° - (3x + 3°)
m/AEB = 87° - 3x
Now we can set the two expressions for m/AEB equal to each other:
87° - 3x = 174° - 16x
Solving for x, we get:
13x = 87°
x = 6.7°
Therefore, the value of x is 6.7°.