Question

In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A = \angle D\). For \(\triangle ABC \sim \triangle DEF\) by the SAS similarity criterion, the additional condition required is

• A. AB/DE = BC/EF
• B. AB/DE = AC/DF
• C. AB/EF = AC/DF
• D. BC/DF = AC/DE

Hint:

The letters in the similarity name "SAS" show that the angle (A) must be strictly in the middle of the two sides (S).
Always write down the two sides meeting at the vertex of the given angle:
For vertex \(A\): sides are \(AB\) and \(AC\).
For vertex \(D\): sides are \(DE\) and \(DF\).
Match them up as \(\frac{AB}{DE} = \frac{AC}{DF}\).

Answer 1

The Correct Option is B

Step 1: Recall exactly what SAS similarity demands.
SAS similarity needs one pair of equal angles, plus the two sides that form, or include, that angle in each triangle to be in proportion. It is not enough to compare any two sides, they must be the ones touching the equal angle.
Step 2: Identify the sides that include the given angle.
In $\triangle ABC$, $\angle A$ sits between sides $AB$ and $AC$. In $\triangle DEF$, $\angle D$ sits between sides $DE$ and $DF$.
Step 3: Write the proportionality condition.
The sides including the equal angles must be proportional: \[ \frac{AB}{DE} = \frac{AC}{DF} \]
Step 4: Conclude.
This is the additional condition needed alongside $\angle A = \angle D$ for SAS similarity.
\[ \boxed{\dfrac{AB}{DE} = \dfrac{AC}{DF}} \]