Question

the line segments $overline{ad}$ and $overline{bc}$ in the figure are parallel. what is the value of $x$? round the answer to the nearest tenth.

Explanation:

Step1: Use property of parallel lines

Since $\overline{AD}\parallel\overline{BC}$, the alternate - interior angles are equal. In $\triangle ABD$, if we consider the angle formed by the parallel lines, we know that the angle at $B$ in $\triangle ABD$ corresponding to the $40^{\circ}$ angle in the right - angled part is also $40^{\circ}$. And in $\triangle ABD$, the sum of interior angles of a triangle is $180^{\circ}$. The triangle $\triangle ABD$ is isosceles (the two line - segments with cross - marks are equal), so the base angles are equal. Let's assume the base angles are $\angle A=\angle ADB$.
We know that $\angle A = 4x + 30$ and $\angle ADB=4x + 30$, and the third angle (the angle at $B$ in $\triangle ABD$) is $40^{\circ}$.
So, $(4x + 30)+(4x + 30)+40=180$.

Step2: Simplify the equation

First, combine like terms:
\[

$$\begin{align*} (4x+4x)+(30 + 30+40)&=180\\ 8x+100&=180 \end{align*}$$

\]
Then, subtract 100 from both sides of the equation: $8x=180 - 100=80$.

Step3: Solve for $x$

Divide both sides of the equation $8x = 80$ by 8: $x=\frac{80}{8}=10.0$.

Answer:

$10.0$