Question

1. in triangle ( cdb ) with an exterior angle at ( b ) (ray ( bw )). angle at ( c ) is ( 46^circ ), side ( cd ) is ( -1 + 8x ), side ( cb ) is ( 18x + 5 ), and there is a ray ( bw ) from ( b ).

Explanation:

Step1: Identify the exterior angle theorem

In a triangle, an exterior angle is equal to the sum of the two non - adjacent interior angles. Here, $\angle CBW$ (with measure $18x + 5$) is an exterior angle of $\triangle BCD$, and the two non - adjacent interior angles are $\angle C$ (with measure $46^{\circ}$) and $\angle D$ (with measure $-1+8x$). So by the exterior angle theorem, we have the equation:
$18x + 5=46+(-1 + 8x)$

Step2: Solve the equation for x

First, simplify the right - hand side of the equation:
$18x + 5=46 - 1+8x$
$18x + 5=45+8x$

Then, subtract $8x$ from both sides:
$18x-8x + 5=45+8x - 8x$
$10x+5 = 45$

Next, subtract 5 from both sides:
$10x+5 - 5=45 - 5$
$10x=40$

Finally, divide both sides by 10:
$x=\frac{40}{10}=4$

Answer:

$x = 4$