Question

If ABC and XYZ are two similar triangles and \(\angle A = 75^\circ\) and \(\angle Y = 57^\circ\), then the value of \(\angle C\) will be

• A. \( 58^\circ \)
• B. \( 48^\circ \)
• C. \( 45^\circ \)
• D. \( 54^\circ \)

Hint:

Pay close attention to the order of vertices in the similarity statement (\(\triangle ABC \sim \triangle XYZ\)). It tells you exactly which angles and sides correspond. A corresponds to X, B to Y, and C to Z. Misinterpreting this correspondence is a common error.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem is based on the properties of similar triangles. When two triangles are similar, their corresponding angles are equal. The sum of angles in any triangle is always 180°.
Step 2: Key Formula or Approach:
1. For similar triangles \(\triangle ABC \sim \triangle XYZ\), we have \(\angle A = \angle X\), \(\angle B = \angle Y\), and \(\angle C = \angle Z\).
2. The sum of angles in \(\triangle ABC\) is \(\angle A + \angle B + \angle C = 180^\circ\).
Step 3: Detailed Explanation:
We are given that \(\triangle ABC\) is similar to \(\triangle XYZ\).
This means their corresponding angles are equal.
\[ \angle A = \angle X \] \[ \angle B = \angle Y \] \[ \angle C = \angle Z \] We are given the values:
\[ \angle A = 75^\circ \] \[ \angle Y = 57^\circ \] From the property of similar triangles, we know that \(\angle B = \angle Y\).
Therefore, \(\angle B = 57^\circ\).
Now, consider \(\triangle ABC\). The sum of its angles must be 180°.
\[ \angle A + \angle B + \angle C = 180^\circ \] Substitute the known values of \(\angle A\) and \(\angle B\):
\[ 75^\circ + 57^\circ + \angle C = 180^\circ \] \[ 132^\circ + \angle C = 180^\circ \] Subtract 132° from both sides to find \(\angle C\):
\[ \angle C = 180^\circ - 132^\circ \] \[ \angle C = 48^\circ \] Step 4: Final Answer:
The value of \(\angle C\) is \( 48^\circ \).