Problem
GEO-B1-M01-P008 A Line Through a Vertex of a Triangle
Through vertex \(B\) of triangle \(ABC\), a line \(l\parallel AC\) is drawn. On one side of line \(l\), rays \(BA\) and \(BC\) divide the straight angle into three angles in the ratio \(3:8:4\). Find the angles of triangle \(ABC\).
These three angles correspond to angles \(A\), \(B\), and \(C\) of the triangle.
Since \(l\parallel AC\), the two angles at line \(l\) are equal to angles \(A\) and \(C\) of the triangle, while the middle angle is \(\angle B\). Thus the triangle angles are in the ratio \(3:8:4\). Let them be \(3x\), \(8x\), and \(4x\). Then \(15x=180^\circ\), so \(x=12^\circ\). The angles are \(36^\circ\), \(96^\circ\), and \(48^\circ\).
Make sure the student explicitly identifies which angles are transferred by parallel lines.