What is the angle at $B$ in the triangle $ABC$, if the angle made at $K$ is $30^\circ$ and the triangles shown in the figure are similar?
$30^\circ$
To find the angle at \(B\) in triangle \(ABC\), given that the triangles are similar and the angle at \(K\) is \(30^\circ\), we need to analyze the properties of similar triangles.
Step 1: Understand Similarity in Triangles
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. In this case, the similarity of the triangles implies all corresponding angles between the triangles are equal.
Step 2: Identify Corresponding Angles
From the diagram, if \(\triangle KJL\) is similar to \(\triangle ACB\), then:
- \(\angle K = \angle A\)
- \(\angle J = \angle C\)
- \(\angle L = \angle B\)
Since we know \(\angle K = 30^\circ\), this implies that \(\angle A = 30^\circ\).
Step 3: Calculate and Confirm Angle at B
Given that the triangles are similar, the corresponding angles are equal:
Therefore, \(\angle B = \angle L = 30^\circ\).
Conclusion:
The angle at \(B\) in triangle \(ABC\) is \(30^\circ\).
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