Questions: Analyze the worked example. Then, answer the questions. After one reflection, point R on triangle VAR maps onto point of triangle BKF. After two reflections, side VR of triangle VAR maps onto side of triangle BKF. After two reflections, side AR of triangle VAR maps onto side of triangle BKF. After two reflections, side AV of triangle VAR maps onto side of triangle BKF. Any point can be mapped onto another point in the plane in, at most, reflection(s). Any line segment can be mapped onto a congruent line segment in, at most, reflection(s). Because a series of reflections preserves distance and angle measures, triangle VAR is is not congruent to triangle BKF.

Analyze the worked example. Then, answer the questions.

After one reflection, point R on triangle VAR maps onto point of triangle BKF.

After two reflections, side VR of triangle VAR maps onto side of triangle BKF.

After two reflections, side AR of triangle VAR maps onto side of triangle BKF.

After two reflections, side AV of triangle VAR maps onto side of triangle BKF.

Any point can be mapped onto another point in the plane in, at most, reflection(s). Any line segment can be mapped onto a congruent line segment in, at most, reflection(s). Because a series of reflections preserves distance and angle measures, triangle VAR is is not congruent to triangle BKF.

Transcript text: Analyze the worked example. Then, answer the questions. After one reflection, point $R$ on $\triangle V A R$ maps onto point $\square$ of $\triangle B K F$. After two reflections, side $V R$ of $\triangle V A R$ maps onto side $\square$ of $\triangle B K F$. After two reflections, side $A R$ of $\triangle V A R$ maps onto side $\square$ of $\triangle B K F$. After two reflections, side $A V$ of $\triangle V A R$ maps onto side $\square$ of $\triangle B K F$. Any point can be mapped onto another point in the plane in, at most, $\square$ reflection(s). Any line segment can be mapped onto a congruent line segment in, at most, $\square$ reflection(s). Because a series of reflections preserves distance and angle measures, $\triangle V A R$ is is not congruent to $\triangle B K F$.

Solution

Solution Steps

To solve the given problem, we need to understand the concept of reflections and how they map points and sides of one triangle onto another. The problem involves determining the mapping of points and sides after one or two reflections and understanding the properties of congruence through reflections.

Mapping Points and Sides: Identify how points and sides of one triangle map onto another after a given number of reflections.
Congruence through Reflections: Understand that reflections preserve distance and angle measures, which implies congruence between the triangles if all corresponding sides are congruent.

Step 1: Mapping Point $ R $

After one reflection, point $ R $ on triangle $ VAR $ maps onto point $ B $ of triangle $ BKF $. Thus, we have: \[ R \mapsto B \]

Step 2: Mapping Sides After Two Reflections

After two reflections, the sides of triangle $ VAR $ map onto the corresponding sides of triangle $ BKF $ as follows:

Side $ VR $ maps onto side $ BK $: \[ VR \mapsto BK \]
Side $ AR $ maps onto side $ KF $: \[ AR \mapsto KF \]
Side $ AV $ maps onto side $ BF $: \[ AV \mapsto BF \]

Step 3: Maximum Reflections

Any point can be mapped onto another point in, at most, $ 2 $ reflections: \[ \text{Max reflections for point} = 2 \] Any line segment can be mapped onto a congruent line segment in, at most, $ 2 $ reflections: \[ \text{Max reflections for line segment} = 2 \]

Step 4: Congruence Statement

Since a series of reflections preserves distance and angle measures, triangle $ VAR $ is congruent to triangle $ BKF $ if all corresponding sides are congruent. Therefore, we conclude: \[ \text{Triangle } VAR \cong \text{Triangle } BKF \]

Final Answer

$\boxed{\text{Triangle } VAR \cong \text{Triangle } BKF}$

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