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: Understanding the Question:
The topic of this question is Similar Triangles in Geometry.
When two triangles are similar, their corresponding angles are equal, and their corresponding sides are proportional to one another.
The order of the vertices in the similarity statement \(\Delta ABC \sim \Delta EDF\) is extremely important.
This specific order defines the exact one-to-one correspondence between the vertices, angles, and sides of the two triangles.
We need to analyze each of the given mathematical statements and determine which one is not true based on this correspondence.

Step 2: Key Formula or Approach:
From the similarity relation \(\Delta ABC \sim \Delta EDF\), we can establish the following properties:
1. Corresponding Angles are equal:
\[ \angle A = \angle E \] \[ \angle B = \angle D \] \[ \angle C = \angle F \] 2. Corresponding Sides are proportional:
\[ \frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} \] 3. The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides:
\[ \frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED} \] We will check each option against these established mathematical relations.

Step 3: Detailed Explanation:

• Let us analyze Option (A):
The statement is \(\frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED}\).
Since the ratio of the perimeters of similar triangles equals the ratio of any pair of corresponding sides, and \(AB\) corresponds to \(ED\), this statement is true.

• Let us analyze Option (B):
The statement is \(\frac{AB}{ED} = \frac{AC}{EF}\).
From our established proportionality of sides, we know that \(\frac{AB}{ED} = \frac{AC}{EF}\) is correct because \(AB\) corresponds to \(ED\) and \(AC\) corresponds to \(EF\). Thus, this statement is true.

• Let us analyze Option (C):
The statement asserts that \(\angle A = \angle D\) and \(\angle C = \angle F\).
From the vertex correspondence of \(\Delta ABC \sim \Delta EDF\), we know that vertex \(A\) corresponds to vertex \(E\), which means \(\angle A = \angle E\).
Vertex \(B\) corresponds to vertex \(D\), which means \(\angle B = \angle D\).
Therefore, the statement \(\angle A = \angle D\) is incorrect because \(\angle A\) must correspond and be equal to \(\angle E\), not \(\angle D\). Thus, this option is not true.

• Let us analyze Option (D):
Let us assume the ratio of corresponding sides is equal to a constant \(\lambda\):
\[ \frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} = \lambda \] This gives \(AB = \lambda \cdot ED\), \(BC = \lambda \cdot DF\), and \(AC = \lambda \cdot EF\).
Substitute these expressions into the left-hand side of the equation in Option (D):
\[ \frac{AB + BC}{AC} = \frac{\lambda \cdot ED + \lambda \cdot DF}{\lambda \cdot EF} = \frac{\lambda(ED + DF)}{\lambda \cdot EF} = \frac{ED + DF}{EF} \] Since the left-hand side simplifies to match the right-hand side, this statement is mathematically true.


Step 4: Final Answer:
The statement that is not true is Option (C).
Therefore, the correct option is (C).