Question

in the diagram below, $overline{ab}paralleloverline{df}$, $mangle dce = 105^{circ}$ and $mangle cef = 150^{circ}$. find $mangle b$. you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale. answer attempt 3 out of 6 $mangle b=\boxed{75}$ submit answer

Explanation:

Step1: Find the measure of the angle adjacent to $\angle CEF$

The angle adjacent to $\angle CEF$ forms a linear - pair with $\angle CEF$. A linear - pair of angles sums to $180^{\circ}$. Let the adjacent angle be $\angle CEG$. So, $m\angle CEG=180 - 150=30^{\circ}$.

Step2: Find the measure of the third angle in $\triangle DCE$

In $\triangle DCE$, we know that the sum of the interior angles of a triangle is $180^{\circ}$. Given $m\angle DCE = 105^{\circ}$ and we found $m\angle CEG = 30^{\circ}$. Let the third angle in $\triangle DCE$ be $\angle CDE$. Then $m\angle CDE=180-(105 + 30)=45^{\circ}$.

Step3: Use the property of parallel lines

Since $\overline{AB}\parallel\overline{DF}$, $\angle B$ and $\angle CDE$ are alternate - interior angles. Alternate - interior angles formed by parallel lines are congruent. So, $m\angle B = 75^{\circ}$.

Answer:

$75^{\circ}$