Question

It is given that \(\Delta ABC \sim \Delta EDF\). Which of the following is not true ?

• A. \(\frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED}\)
• B. \(\frac{AB}{ED} = \frac{AC}{EF}\)
• C. \(\angle A = \angle D, \angle C = \angle F\)
• D. \(\frac{AB + BC}{AC} = \frac{DE + DF}{EF}\)

Hint:

To avoid visual confusion in similar triangle problems, always write down the vertex mapping explicitly:
Map \(A \leftrightarrow E\), \(B \leftrightarrow D\), and \(C \leftrightarrow F\) immediately.
This simple step prevents mistakes when identifying corresponding angles and side ratios!

Answer 1

The Correct Option is C

Step 1: Pick a real triangle that fits the similarity.
Since \(\Delta ABC \sim \Delta EDF\) with the vertex order A to E, B to D, C to F, take a concrete right triangle: let \(ED = 3\), \(DF = 4\), \(EF = 5\), right angled at D. Then \(AB = 3\), \(BC = 4\), \(AC = 5\), right angled at B, so the two triangles are genuinely similar with scale factor 1.
Step 2: Check the true angle correspondence.
Because B corresponds to D, the right angle is at B and also at D, so \(\angle B = \angle D = 90^\circ\). Angle A corresponds to angle E, not angle D, and here \(\angle A \approx 53.13^\circ\) while \(\angle D = 90^\circ\), so they are clearly not equal.
Step 3: Test each option with these numbers.
Option (A): Perimeter ratio \(= \frac{12}{12} = 1\), and \(\frac{AB}{ED} = \frac{3}{3} = 1\), true. Option (B): \(\frac{AB}{ED} = 1\) and \(\frac{AC}{EF} = \frac{5}{5} = 1\), true. Option (D): \(\frac{AB+BC}{AC} = \frac{7}{5}\) and \(\frac{DE+DF}{EF} = \frac{7}{5}\), true. Option (C) claims \(\angle A = \angle D\), but we just showed \(\angle A \ne \angle D\), so this is the false statement.
Step 4: Conclude.
The statement that is not true is option (C), matching the given answer.
\[ \boxed{\text{Option (C)}} \]