Question

determine the rigid transformations that will map (\triangle abc) to (\triangle xyz).
translate vertex b to vertex z; reflect (\triangle abc) across side ab.
translate vertex x to vertex c; rotate (\triangle xyz) to align the sides and angles.
reflect (\triangle abc) across side ab; translate vertex c to vertex x.
translate vertex x to vertex a; rotate (\triangle xyz) to align the sides and angles.

Explanation:

Step1: Analyze the first option

Translate vertex \(B\) to vertex \(Z\); reflect \(\triangle ABC\) across side \(AB\). This does not map the triangles correctly as the orientation and side - angle relationships are not maintained.

Step2: Analyze the second option

Translate vertex \(X\) to vertex \(C\); rotate \(\triangle XYZ\) to align the sides and angles. This is incorrect as the starting translation is not in the right direction for mapping.

Step3: Analyze the third option

Reflect \(\triangle ABC\) across side \(AB\); translate vertex \(C\) to vertex \(X\).
First, when we reflect \(\triangle ABC\) across \(AB\), we start to re - orient the triangle. Then, translating vertex \(C\) (after reflection) to vertex \(X\) will map the sides and angles of \(\triangle ABC\) to \(\triangle XYZ\) because the side lengths (since \(AC\) is equal to the corresponding side in \(\triangle XYZ\) as per the triangle congruence conditions from angle - side relationships (\(A = 35^{\circ}\), \(B=98^{\circ}\), so \(C = 180-(35 + 98)=47^{\circ}\) and in \(\triangle XYZ\) \(Z = 47^{\circ}\), \(X = 35^{\circ}\)) and side - angle - side (or angle - side - angle) congruence can be established after the rigid transformations.

Step4: Analyze the fourth option

Translate vertex \(X\) to vertex \(A\); rotate \(\triangle XYZ\) to align the sides and angles. This is incorrect as the translation is not the right initial move for mapping.

Answer:

Reflect \(\triangle ABC\) across side \(AB\); translate vertex \(C\) to vertex \(X\).