Question

which of the following would allow you to conclude that △abc is similar to △edc? a. $overline{ab}congoverline{de}$ b. $overleftrightarrow{ab}$ is parallel to $overleftrightarrow{de}$ c. $overline{ab}congoverline{de}$ and $overline{ac}congoverline{dc}$ d. $overleftrightarrow{ab}$ is perpendicular to $overleftrightarrow{bd}$ e. $mangle bca = 30^{circ}$ and $mangle ced = 60^{circ}$

Explanation:

Step1: Recall similarity - criteria

Similar triangles can be proven by AA (angle - angle), SAS (side - angle - side), SSS (side - side - side) etc. If two lines are parallel, corresponding angles are equal.

Step2: Analyze each option

• Option A: $\overline{AB}\cong\overline{DE}$ only gives information about one pair of sides being congruent, not enough for similarity.
• Option B: If $\overleftrightarrow{AB}\parallel\overleftrightarrow{DE}$, then $\angle BAC=\angle DEC$ and $\angle ABC=\angle EDC$ (corresponding angles). By the AA (angle - angle) similarity criterion, $\triangle ABC\sim\triangle EDC$.
• Option C: $\overline{AB}\cong\overline{DE}$ and $\overline{AC}\cong\overline{DC}$ gives information about side - side, but not in the correct proportion or with an included angle for SAS similarity.
• Option D: $\overleftrightarrow{AB}\perp\overleftrightarrow{BD}$ gives no information about the angles or sides of the two triangles for similarity.
• Option E: Just knowing $m\angle BCA = 30^{\circ}$ and $m\angle CED=60^{\circ}$ is not enough to prove similarity as we don't know other angle relationships.

Answer:

B. $\overleftrightarrow{AB}$ is parallel to $\overleftrightarrow{DE}$