Question

keisha, dwayne, and lonzell are planning for a new bridge to replace the old bridge. the new bridge will start at point b, where dwayne is standing, and end at point c, where keisha is standing. lonzell walks to point d and then walks parallel to the river until he reaches point e where he sees dwayne and keisha are aligned. why is the distance from e to b the length of the new bridge? it is given that because they are vertical angles. because they are alternate interior angles. therefore, △bac is congruent to △bde by . so, (overline{cb}congoverline{be}) because the corresponding parts of congruent triangles are congruent. this means that the distance from e to b is the length of the new bridge.

Explanation:

Step1: Identify vertical - angle pair

$\angle BAC=\angle BDE$ because they are vertical angles.

Step2: Identify alternate - interior angle pair

$\angle ABC=\angle BED$ because they are alternate interior angles.

Step3: Determine triangle - congruence criterion

$\triangle BAC$ is congruent to $\triangle BDE$ by the Angle - Angle (AA) similarity criterion (in the case of right - angled triangles here, AA can be used for congruence as well since we have two pairs of equal angles and the right - angle is common in a sense of the perpendicularity of the river - banks).

Answer:

It is given that $\angle BAC=\angle BDE$ because they are vertical angles. $\angle ABC=\angle BED$ because they are alternate interior angles. Therefore, $\triangle BAC$ is congruent to $\triangle BDE$ by AA (Angle - Angle). So, $\overline{CB}\cong\overline{BE}$ because the corresponding parts of congruent triangles are congruent. This means that the distance from E to B is the length of the new bridge.