Question

proving congruent triangles using transformations
determine the rigid transformations that will map △abc to △xyz.
four option cards:

1. translate vertex b to vertex z; reflect △abc across side ab.
2. translate vertex x to vertex a; rotate △xyz to align the sides and angles.
3. translate vertex x to vertex c; rotate △xyz to align the sides and angles.
4. reflect △abc across side ab; translate vertex c to vertex x.

triangles △abc and △xyz with angle and side markings: △abc has angles 35°, 98°, and side ac marked; △xyz has angles 35°, 47°, and a side marked.

Explanation:

Step1: Analyze Triangle Angles

First, find the third angle of each triangle. For $\triangle ABC$: $180^\circ - 35^\circ - 98^\circ = 47^\circ$. For $\triangle XYZ$: $180^\circ - 35^\circ - 47^\circ = 98^\circ$. So angles are $35^\circ, 98^\circ, 47^\circ$ in both, and a side with one mark (equal length).

Step2: Match Vertices

$\angle A = \angle X = 35^\circ$, $\angle B = \angle Z = 98^\circ$ (wait, no, $\angle Z = 47^\circ$, $\angle B = 98^\circ$—wait, recalculate: $\triangle ABC$ angles: $A=35^\circ$, $B=98^\circ$, $C=47^\circ$. $\triangle XYZ$: $X=35^\circ$, $Z=47^\circ$, $Y=98^\circ$ (since $180 - 35 - 47 = 98$). So vertex $A$ (35°) matches $X$ (35°), vertex $B$ (98°) matches $Y$ (98°), vertex $C$ (47°) matches $Z$ (47°).

Step3: Analyze Transformations

We need to map $A$ to $X$, $B$ to $Y$, $C$ to $Z$. Let's check the options:

Option 1: Translate vertex $X$ to $C$; rotate $\triangle XYZ$ to align. No, direction wrong.

Option 2: Translate $B$ to $Z$; reflect $\triangle ABC$ over $AB$. Not matching.

Option 3: Reflect $\triangle ABC$ over $AB$; translate $C$ to $X$. No.

Option 4: Translate vertex $X$ to $A$; rotate $\triangle XYZ$ to align sides/angles. Wait, the correct transformation: Translate vertex $X$ to vertex $A$ (since $A$ and $X$ both 35°), then rotate $\triangle XYZ$ so that the sides (with the mark) and angles align. Wait, the option "Translate vertex X to vertex A; rotate $\triangle XYZ$ to align the sides and angles"—wait, the options as per the cards (let's parse the cards):

Card 4 (bottom right? Wait the cards:

1. Translate vertex X to C; rotate XYZ to align (no, X to C? X is 35°, C is 47°—no).

1. Translate B to Z; reflect ABC over AB (B to Z? B is 98°, Z is 47°—no).

1. Reflect ABC over AB; translate C to X (C is 47°, X is 35°—no).

1. Translate X to A; rotate XYZ to align (X (35°) to A (35°), then rotate so that the 98° and 47° angles match, and the marked side (equal length) aligns. This makes sense because after translating X to A, rotating will align the other vertices (Y to B, Z to C? Wait no, earlier angle mix-up: $\triangle ABC$: $A=35$, $B=98$, $C=47$; $\triangle XYZ$: $X=35$, $Y=98$, $Z=47$ (since $X=35$, $Z=47$, so $Y=180-35-47=98$). So $X$ (35°) → $A$ (35°), $Y$ (98°) → $B$ (98°), $Z$ (47°) → $C$ (47°). So translating $X$ to $A$, then rotating $\triangle XYZ$ so that $Y$ maps to $B$ and $Z$ maps to $C$, aligning the sides (the marked side in $\triangle XYZ$ (let's say $XY$) with $AB$? Wait, the marked side: in $\triangle ABC$, side $AC$ has one mark? Wait the diagram: $\triangle ABC$ has $AC$ with one mark, $\triangle XYZ$ has $XZ$? No, the marked side in $\triangle XYZ$ is on the side from $X$ to the other vertex (maybe $XY$? Wait the diagram: $\triangle ABC$: $AC$ has one mark. $\triangle XYZ$: the side from $X$ to (let's say $Y$? No, the red mark is on a side of $\triangle XYZ$—probably $XY$ or $XZ$. Anyway, the key is that translating $X$ (35°) to $A$ (35°), then rotating to align the other angles (98° and 47°) and the equal side. So the correct option is "Translate vertex X to vertex A; rotate $\triangle XYZ$ to align the sides and angles" (the fourth option, as per the cards: "Translate vertex X to vertex A; rotate $\triangle XYZ$ to align the sides and angles").

Answer:

The correct transformation is "Translate vertex X to vertex A; rotate $\triangle XYZ$ to align the sides and angles" (the option with this description, likely the bottom - right card among the four, but based on the analysis, the transformation that maps $X$ (35°) to $A$ (35°) and then rotates to align the 98° and 47° angles and equal - length side).