Question

triangles abc and dec are shown. given $overline{ab}$ is parallel to $overline{de}$, which statement about $\triangle abc$ and $\triangle dec$ is true?
$ab = de$
$mangle abe=mangle deb$
$mangle dab=mangle deb$
$ac = dc$

Explanation:

Step1: Recall parallel - line properties

Since $\overline{AB}\parallel\overline{DE}$, alternate - interior angles are equal. $\angle ABE$ and $\angle DEB$ are alternate - interior angles formed by the parallel lines $\overline{AB}$ and $\overline{DE}$ and the transversal $\overline{BE}$.

Step2: Analyze each option

• Option 1: Just because $\overline{AB}\parallel\overline{DE}$, we cannot say $AB = DE$ without more information about the triangles.
• Option 2: $\angle ABE$ and $\angle DEB$ are alternate - interior angles. By the property of parallel lines, $m\angle ABE=m\angle DEB$.
• Option 3: $\angle DAB$ and $\angle DEB$ are not related in a way that they are equal just because $\overline{AB}\parallel\overline{DE}$.
• Option 4: There is no information from the parallel lines $\overline{AB}\parallel\overline{DE}$ that would allow us to conclude $AC = DC$.

Answer:

$m\angle ABE=m\angle DEB$