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: Understanding the Question:
This question is from Geometry, specifically about similarity criteria for triangles.
We need to determine the required condition for two triangles to be similar via the Side-Angle-Side (SAS) Similarity Criterion, given that one pair of angles is equal.

Step 2: Key Formula or Approach:
The SAS (Side-Angle-Side) Similarity Criterion states:
If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar.
The key phrase is "sides including these angles". This means the two sides that form the given equal angle.

Step 3: Detailed Explanation:
We are given two triangles, \(\triangle ABC\) and \(\triangle DEF\), with \(\angle A = \angle D\).
Let us identify the sides that form these angles in each triangle:
In \(\triangle ABC\), \(\angle A\) is included between the sides \(AB\) and \(AC\).
In \(\triangle DEF\), \(\angle D\) is included between the sides \(DE\) and \(DF\).
For the triangles to be similar under the SAS criterion, the ratio of these including sides must be equal:
\[ \frac{\text{Side containing } \angle A \text{ in } \triangle ABC}{\text{Corresponding side containing } \angle D \text{ in } \triangle DEF} \]
This translates to:
\[ \frac{AB}{DE} = \frac{AC}{DF} \]
Thus, this is the precise condition needed to satisfy the SAS similarity criterion.

Step 4: Final Answer:
The additional condition required is AB/DE = AC/DF.