Math worksheet titled "Rigid Transformations and Corresponding Parts" featuring six problems with geometric figures on coordinate planes to be transformed using translations, rotations, and reflections.

Let's solve each of the six transformation problems step by step. Since I can't directly see or draw on the image, I'll guide you through how to perform each transformation, including:

- Identifying coordinates of original points.
- Applying the transformation rule.
- Plotting the image.
- Labeling corresponding sides and angles.

We’ll go one by one.

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1) Translation: 4 units left and 6 units down

Original triangle: $ \triangle LMK $

From the graph:
- $ L = (3, 5) $
- $ M = (0, 2) $
- $ K = (4, 1) $

Translation Rule:
Move each point:
$ (x, y) \rightarrow (x - 4, y - 6) $

Apply:
- $ L' = (3 - 4, 5 - 6) = (-1, -1) $
- $ M' = (0 - 4, 2 - 6) = (-4, -4) $
- $ K' = (4 - 4, 1 - 6) = (0, -5) $

Plot: Draw triangle $ \triangle L'M'K' $ with these new points.

Labeling:
- $ L \leftrightarrow L' $, $ M \leftrightarrow M' $, $ K \leftrightarrow K' $
- Sides: $ LM \leftrightarrow L'M' $, $ MK \leftrightarrow M'K' $, $ KL \leftrightarrow K'L' $
- Angles: $ \angle L \leftrightarrow \angle L' $, etc.

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2) Translation: 1 unit left and 5 units down

Original quadrilateral: $ \triangle RSTU $

From the graph:
- $ R = (2, 1) $
- $ S = (2, 4) $
- $ T = (5, 5) $
- $ U = (5, 1) $

Translation Rule:
$ (x, y) \rightarrow (x - 1, y - 5) $

Apply:
- $ R' = (1, -4) $
- $ S' = (1, -1) $
- $ T' = (4, 0) $
- $ U' = (4, -4) $

Plot: Draw quadrilateral $ R'S'T'U' $

Labeling:
- $ R \leftrightarrow R' $, $ S \leftrightarrow S' $, etc.
- Sides: $ RS \leftrightarrow R'S' $, $ ST \leftrightarrow S'T' $, $ TU \leftrightarrow T'U' $, $ UR \leftrightarrow U'R' $
- Angles: Corresponding angles match.

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3) Rotation 90° counterclockwise about the origin

Original quadrilateral: $ \triangle WXYZ $

Points:
- $ W = (0, 0) $
- $ X = (0, 3) $
- $ Y = (1, 4) $
- $ Z = (3, 4) $

Rotation Rule (90° CCW about origin):
$ (x, y) \rightarrow (-y, x) $

Apply:
- $ W' = (0, 0) $
- $ X' = (-3, 0) $
- $ Y' = (-4, 1) $
- $ Z' = (-4, 3) $

Plot: Draw $ W'X'Y'Z' $

Labeling:
- $ W \leftrightarrow W' $, $ X \leftrightarrow X' $, etc.
- Sides: $ WX \leftrightarrow W'X' $, $ XY \leftrightarrow X'Y' $, etc.
- Angles preserved.

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4) Reflection across the y-axis

Original triangle: $ \triangle PQR $

Points:
- $ P = (2, 1) $
- $ Q = (4, 3) $
- $ R = (5, -1) $

Reflection over y-axis:
$ (x, y) \rightarrow (-x, y) $

Apply:
- $ P' = (-2, 1) $
- $ Q' = (-4, 3) $
- $ R' = (-5, -1) $

Plot: Reflect triangle across y-axis.

Labeling:
- $ P \leftrightarrow P' $, $ Q \leftrightarrow Q' $, $ R \leftrightarrow R' $
- Sides: $ PQ \leftrightarrow P'Q' $, $ QR \leftrightarrow Q'R' $, $ RP \leftrightarrow R'P' $
- Angles remain equal.

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5) Reflection across the x-axis

Original triangle: $ \triangle JKL $

Points:
- $ J = (-2, -1) $
- $ K = (-4, 1) $
- $ L = (-1, 3) $

Reflection over x-axis:
$ (x, y) \rightarrow (x, -y) $

Apply:
- $ J' = (-2, 1) $
- $ K' = (-4, -1) $
- $ L' = (-1, -3) $

Plot: Reflect triangle across x-axis.

Labeling:
- $ J \leftrightarrow J' $, $ K \leftrightarrow K' $, $ L \leftrightarrow L' $
- Sides: $ JK \leftrightarrow J'K' $, $ KL \leftrightarrow K'L' $, $ LJ \leftrightarrow L'J' $
- Angles are preserved.

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6) Rotation 90° counterclockwise about the origin

Original triangle: $ \triangle DEF $

Points:
- $ D = (1, 1) $
- $ E = (3, 4) $
- $ F = (4, 1) $

Rule: $ (x, y) \rightarrow (-y, x) $

Apply:
- $ D' = (-1, 1) $
- $ E' = (-4, 3) $
- $ F' = (-1, 4) $

Plot: Draw $ \triangle D'E'F' $

Labeling:
- $ D \leftrightarrow D' $, $ E \leftrightarrow E' $, $ F \leftrightarrow F' $
- Sides: $ DE \leftrightarrow D'E' $, $ EF \leftrightarrow E'F' $, $ FD \leftrightarrow F'D' $
- Angles preserved.

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✔ Summary of Key Concepts

| Transformation | Rule |
|---------------|------|
| Translation (a,b) | $ (x,y) \rightarrow (x+a, y+b) $ |
| Reflection over y-axis | $ (x,y) \rightarrow (-x,y) $ |
| Reflection over x-axis | $ (x,y) \rightarrow (x,-y) $ |
| Rotation 90° CCW about origin | $ (x,y) \rightarrow (-y,x) $ |

> 🔁 All transformations are rigid motions, so:
> - Lengths of sides are preserved.
> - Measures of angles are preserved.
> - Shapes are congruent.

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📌 Final Steps for Student:

For each problem:
1. Identify the original coordinates.
2. Apply the transformation rule.
3. Plot the new points.
4. Connect them in the same order.
5. Label corresponding vertices with primes (e.g., $ A \to A' $).
6. Use tick marks or labels to show corresponding sides and angles.

If you'd like, I can provide a downloadable version of this solution with labeled graphs (as text-based coordinate plots), or help you sketch it on paper. Let me know!