Question

what else would need to be congruent to show that △ abc ≅ △ def by asa?

a. ∠c ≅ ∠f
b. overline{ac} ≅ overline{de}
c. ∠a ≅ ∠d
d. overline{bc} ≅ overline{ef}

Explanation:

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Step2: Analyze Given Angles

In $\triangle ABC$ and $\triangle DEF$:

• $\angle B = 50^{\circ}$ and $\angle E=50^{\circ}$, so $\angle B\cong\angle E$.
• $\angle C = 45^{\circ}$ and $\angle F = 45^{\circ}$, so $\angle C\cong\angle F$.

For ASA, we need the side included between the two angles. In $\triangle ABC$, the side included between $\angle B$ and $\angle C$ is $BC$. In $\triangle DEF$, the side included between $\angle E$ and $\angle F$ is $EF$. So we need $BC\cong EF$ to apply ASA.

Let's analyze other options:

• Option A: $\angle C\cong\angle F$ is already true (both $45^{\circ}$), so it's not what we need.
• Option B: $\overline{AC}\cong\overline{DE}$ - $AC$ is not the included side between the two angles for ASA in this case.
• Option C: $\angle A\cong\angle D$ - We can calculate $\angle A=180 - 50 - 45=85^{\circ}$ and $\angle D = 180 - 50 - 45 = 85^{\circ}$, but this is not required for ASA as we already have two angles and need the included side.

Answer:

D. $\overline{BC}\cong\overline{EF}$