Answer:
Explanation:
To determine whether AB is the line of least common denominator (LCD) for CD and EF, we need to examine the given information and apply relevant geometric principles.
Given:
CD || EF (CD is parallel to EF)
AB intersects CD at P and EF at Q
m/APC = 4x + 18
m/EQB = 6x - 18
To determine if AB is the LCD, we need to check if the corresponding angles formed by CD and EF with AB are proportional.
First, let's consider the angle formed by CD and AB at point P, denoted as angle APC. According to the given information, m/APC = 4x + 18.
Next, let's consider the angle formed by EF and AB at point Q, denoted as angle EQB. According to the given information, m/EQB = 6x - 18.
For AB to be the LCD, the corresponding angles APC and EQB must be proportional. In other words, their angle measures must be equal up to a constant factor.
Therefore, we can set up an equation to compare the two angles:
4x + 18 = k(6x - 18)
where k is the constant of proportionality.
Now, let's solve this equation to find the value of x:
4x + 18 = 6kx - 18k
18k + 18 = 6kx - 4x
18k + 18 = (6k - 4)x
18k + 18 = 2(3k - 2)x
For the equation to hold true for all values of x, the coefficients on both sides of the equation must be equal. Therefore, we have:
18k + 18 = 0
Solving this equation for k, we get:
k = -1
Substituting this value of k back into the equation, we have:
4x + 18 = -1(6x - 18)
4x + 18 = -6x + 18
10x = 0
x = 0
Now, let's determine the angle measures of APC and EQB using the value of x:
m/APC = 4x + 18 = 4(0) + 18 = 18
m/EQB = 6x - 18 = 6(0) - 18 = -18
The angle measures of APC and EQB are 18 and -18, respectively. Since these angles are not equal, AB is not the LCD for CD and EF.
In conclusion, based on the given information, AB is not the line of least common denominator (LCD) for CD and EF because the corresponding angles formed by CD and EF with AB are not proportional.