Question

there is a diagram with two vertical lines (parallel lines) and a transversal line. the angles formed are labeled with expressions and numbers: one angle is (3x + 169), another is (-4x - 7), and there are numbered angles 1, 3, 4, 6, 7, 8.

Explanation:

Step1: Identify congruent angles

Since the two vertical lines are parallel, the corresponding angles $3x + 169$ and $-4x - 7$ are supplementary (they form a linear pair with congruent angles, so they add up to $180^\circ$).
$$3x + 169 + (-4x - 7) = 180$$

Step2: Simplify the equation

Combine like terms in the equation.
$$3x - 4x + 169 - 7 = 180$$
$$-x + 162 = 180$$

Step3: Solve for $x$

Isolate the variable $x$ by subtracting 162 from both sides, then multiply by -1.
$$-x = 180 - 162$$
$$-x = 18$$
$$x = -18$$

Step4: Calculate angle measures (optional verification)

Substitute $x=-18$ into $3x+169$:
$$3(-18)+169 = -54 + 169 = 115$$
Substitute $x=-18$ into $-4x-7$:
$$-4(-18)-7 = 72 - 7 = 65$$
Confirm $115 + 65 = 180$, which is correct.

Answer:

$x = -18$