Question

question 8 of 10
based only on the information given in the diagram, it is guaranteed that
△abc ~ △def.

a. true
b. false

Explanation:

Step1: Recall Similar Triangle Criteria

Similar triangles require AA (Angle - Angle), SAS (Side - Angle - Side), or SSS (Side - Side - Side) similarity. For right triangles, if we have two right angles (so one pair of equal angles) and the sides around the right angle are in proportion, then SAS similarity applies.

Step2: Analyze Given Triangles

• $\triangle ABC$: Right - angled at $C$, with $AC = 14$.
• $\triangle DEF$: Right - angled at $F$, with $DF=7$.

We know that $\angle C=\angle F = 90^{\circ}$, so one pair of angles is equal. But we only know the length of one leg of each triangle. We don't know anything about the other legs or the hypotenuses, nor do we know if the other angles are equal. For SAS similarity, we need the ratio of the two corresponding sides around the equal angle to be equal. Here, we only have one side from each triangle, so we can't confirm that the ratio of the sides (e.g., $\frac{AC}{DF}$ and the ratio of the other legs) is equal. Also, we don't have AA (since we only know one pair of angles is equal, we need another pair) or SSS. So we can't guarantee that $\triangle ABC\sim\triangle DEF$.

Answer:

B. False