Question

how can δabc be mapped to δxyz? first, translate ______. next, rotate δabc about b to align the sides and angles. vertex b to vertex z vertex b to vertex y vertex a to vertex z vertex a to vertex y

Explanation:

Step1: Analyze the correspondence of vertices

In congruent triangles (assuming \(\triangle ABC\) and \(\triangle XYZ\) are congruent as per side - angle - side ( \(AB = XY=25\mathrm{cm}\), \(\angle A=\angle X = 36^{\circ}\), \(AC=XZ = 29\mathrm{cm}\))), we need to match the vertices.
The side \(AB\) in \(\triangle ABC\) corresponds to the side \(XY\) in \(\triangle XYZ\). The vertex \(A\) in \(\triangle ABC\) should be mapped to the vertex \(X\) (but there is no such option), and if we consider the order of the triangles \(\triangle ABC\) and \(\triangle XYZ\) and the given rotation about \(B\) (in \(\triangle ABC\)), we note that the side \(AB\) (with length \(25\mathrm{cm}\)) and \(XY\) (with length \(25\mathrm{cm}\)) are corresponding. So, to start the mapping (translation part), we should translate vertex \(A\) (which is an endpoint of the side with length \(25\mathrm{cm}\) in \(\triangle ABC\)) to vertex \(Y\) (an endpoint of the side with length \(25\mathrm{cm}\) in \(\triangle XYZ\)). Then, a rotation about \(B\) (after translation) can align the sides and angles.
If we consider the other options:

• Translating vertex \(B\) to \(Z\) or \(Y\) first is not correct because the side - angle - side correspondence starts with the side adjacent to the given angle. For example, in \(\triangle ABC\) the side \(AB\) and in \(\triangle XYZ\) the side \(XY\) are the sides adjacent to the \(36^{\circ}\) angles.
• Translating vertex \(A\) to \(Z\) is wrong as \(AB\) (\(25\mathrm{cm}\)) corresponds to \(XY\) (\(25\mathrm{cm}\)) not \(XZ\) (\(29\mathrm{cm}\)).

Answer:

vertex A to vertex Y