Question

which of these triangle pairs can be mapped to each other using both a translation and a reflection across the line containing \\( \overline { a b } \\)?

Explanation:

Step1: Analyze translation and reflection

Translation moves a figure without rotation or reflection. Reflection flips a figure over a line.

Step2: Check first triangle pair

For the first pair (with \(X,Y,Z\) and \(A,B,C\)), we can translate \(\triangle XYZ\) along the line \(XY\) (which is colinear with \(AB\)) and then reflect over line \(AB\) to map it to \(\triangle ABC\) as the sides and angles match for congruent triangles (by SSS - side - side - side as marked).

Step3: Check second triangle pair

The second pair (with \(A,B,C,Y\)) does not have a clear translation - reflection mapping as the orientation and side - angle - side (if we assume markings) does not align for a combined translation - reflection over \(AB\) mapping.

Answer:

The first triangle pair (with \(X,Y,Z\) and \(A,B,C\)) can be mapped using both a translation and a reflection across the line containing \(\overline{AB}\).