Question

the angle measurements in the diagram are represented by the following expressions.
$angle a = 6x - 18^{circ}$
$angle b = 14x + 38^{circ}$

Explanation:

Step1: Identify angle - relationship

Assume ∠A and ∠B are corresponding angles (since the lines are parallel as indicated by the arrow - heads), so ∠A = ∠B.
$$6x - 18=14x + 38$$

Step2: Solve the equation for x

First, move the x - terms to one side and the constants to the other side.
$$6x-14x=38 + 18$$
$$- 8x=56$$
Then, divide both sides by - 8.
$$x=\frac{56}{-8}=-7$$

Step3: Find the measure of ∠A

Substitute x = - 7 into the expression for ∠A.
$$\angle A=6x-18=6\times(-7)-18=-42 - 18=-60$$
Since angle measures cannot be negative in this context, we made a wrong assumption. Let's assume ∠A and ∠B are supplementary (adjacent angles on a straight - line formed by the transversal with the parallel lines).
So, ∠A+∠B = 180°.
$$(6x - 18)+(14x + 38)=180$$

Step4: Simplify the left - hand side of the equation

Combine like terms.
$$6x+14x-18 + 38=180$$
$$20x+20 = 180$$

Step5: Solve for x

Subtract 20 from both sides.
$$20x=180 - 20=160$$
Divide both sides by 20.
$$x=\frac{160}{20}=8$$

Step6: Find the measure of ∠A

Substitute x = 8 into the expression for ∠A.
$$\angle A=6x-18=6\times8-18=48 - 18=30^{\circ}$$

Answer:

30°