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 conclude that $AB = DE$ without more information about the triangles' side - length relationships.
• 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 by any parallel - line angle property. They are not corresponding, alternate - interior, or alternate - exterior angles.
• Option 4: There is no information from the parallel lines $\overline{AB}$ and $\overline{DE}$ that would imply $AC = DC$.

Answer:

$m\angle ABE=m\angle DEB$