Question

If \( \triangle ABC \cong \triangle ACB \), then \( \triangle ABC \) is Isosceles with :

• A. AB=AC
• B. AB=BC
• C. AC=BC
• D. None of these

Hint:

The order of vertices in a triangle congruence statement is very important. \( \triangle \mathbf{A}\mathbf{B}\mathbf{C} \cong \triangle \mathbf{A}\mathbf{C}\mathbf{B} \) This means:
1st vertex corresponds to 1st: A \(\leftrightarrow\) A
2nd vertex corresponds to 2nd: B \(\leftrightarrow\) C
3rd vertex corresponds to 3rd: C \(\leftrightarrow\) B Corresponding sides are equal:
AB (1st \& 2nd) = AC (1st \& 2nd)
BC (2nd \& 3rd) = CB (2nd \& 3rd)
AC (1st \& 3rd) = AB (1st \& 3rd) From this, \( AB = AC \), so the triangle is isosceles with these two sides equal.

Answer 1

The Correct Option is A

Concept: Congruence of triangles (\(\cong\)) means that two triangles have exactly the same size and shape. The order of vertices in the congruence statement is crucial as it indicates corresponding parts. Step 1: Understanding the congruence statement We are given \( \triangle ABC \cong \triangle ACB \). This statement implies the following correspondence between vertices:
Vertex A in \(\triangle ABC\) corresponds to Vertex A in \(\triangle ACB\).
Vertex B in \(\triangle ABC\) corresponds to Vertex C in \(\triangle ACB\).
Vertex C in \(\triangle ABC\) corresponds to Vertex B in \(\triangle ACB\). Step 2: Identifying corresponding sides Since corresponding parts of congruent triangles are equal (CPCTC), we can equate the lengths of corresponding sides:
Side AB (from \(\triangle ABC\)) corresponds to Side AC (from \(\triangle ACB\)). Therefore, \( AB = AC \).
Side BC (from \(\triangle ABC\)) corresponds to Side CB (from \(\triangle ACB\)). Therefore, \( BC = CB \) (This is trivial, as it's the same side).
Side AC (from \(\triangle ABC\)) corresponds to Side AB (from \(\triangle ACB\)). Therefore, \( AC = AB \) (This is the same as the first conclusion). Step 3: Conclusion about the triangle From the correspondence, we found that \( AB = AC \). A triangle in which at least two sides are equal in length is called an isosceles triangle. Since \( AB = AC \), \(\triangle ABC\) is an isosceles triangle with these two sides being equal. Step 4: Analyzing the options
(1) AB=AC: This matches our finding from the congruence.
(2) AB=BC: This is not necessarily true from the given congruence.
(3) AC=BC: This is not necessarily true from the given congruence.
(4) None of these: Incorrect, as option (1) is correct. Therefore, if \( \triangle ABC \cong \triangle ACB \), then \(\triangle ABC\) is isosceles with \(AB=AC\).