Question

in the drawing below, △ abc is similar to △ def. which choice shows the corresponding sides as equal ratios? options: \\(\frac{ab}{de} = \frac{bc}{ef} = \frac{ac}{df}\\), \\(\frac{ab}{de} = \frac{bc}{ef} = \frac{df}{ac}\\), \\(\frac{df}{ab} = \frac{bc}{ef} = \frac{ac}{de}\\), \\(\frac{ab}{de} = \frac{df}{bc} = \frac{ac}{ef}\\)

Explanation:

Step1: Identify Corresponding Angles

In similar triangles, corresponding angles are equal. For $\triangle ABC$ and $\triangle DEF$:

• $\angle A = \angle D = 80^\circ$, so side $BC$ (opposite $\angle A$) corresponds to side $EF$ (opposite $\angle D$).
• $\angle B = \angle E = 60^\circ$, so side $AC$ (opposite $\angle B$) corresponds to side $DF$ (opposite $\angle E$).
• $\angle C = \angle F = 40^\circ$, so side $AB$ (opposite $\angle C$) corresponds to side $DE$ (opposite $\angle F$).

Step2: Determine Corresponding Sides Ratio

By the property of similar triangles, the ratio of corresponding sides is equal. So:

• Corresponding sides: $AB$ corresponds to $DE$, $BC$ corresponds to $EF$, $AC$ corresponds to $DF$.
• Thus, the ratio is $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$.

Answer:

$\boldsymbol{\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}}$ (the first option)