Lesson 9-1: Congruence and Transformations

Property

A rigid transformation is a change in the position of a figure that perfectly preserves its size and shape. Two figures are congruent ($\cong$) if and only if one can be mapped exactly onto the other by a sequence of one or more rigid transformations:

1. Translations (slides)
2. Reflections (flips)
3. Rotations (turns)

Examples

• Translation (Slide): A triangle $\Delta ABC$ is moved 5 units to the right and 2 units up to map perfectly onto $\Delta A'B'C'$.
• Reflection (Flip): A triangle $\Delta DEF$ is flipped across the y-axis to create a congruent mirror image, $\Delta D'E'F'$.
• Sequence of Transformations: Pentagon $DEFGH$ is translated 2 units up, then rotated 270° clockwise to produce a congruent pentagon $D''E''F''G''H''$.

Explanation

Rigid transformations (also known as isometries) are simply the "vehicles" we use to drive one shape over to park exactly on top of its clone. Think of it as sliding, flipping, or turning a paper cutout on a desk; the cutout itself remains unchanged. By tracking these movements using prime notation ($A \rightarrow A' \rightarrow A''$), we can prove two shapes are identical without measuring them.

Property

A sequence of transformations, or a composition, maps a preimage figure $F$ to a final image figure $F''$. This can be represented as $F'' = (T_2 \circ T_1)(F)$, where transformation $T_1$ is applied first, followed by $T_2$. For any given preimage and image, there can be multiple different sequences of transformations that produce the same result.

Examples

A flip is a reflection. It is a transformation that flips a figure across a line, called the line of reflection, to create a mirror image. The orientation is reversed.

The letter 'p' becomes the letter 'q' when it is reflected across a vertical line.
Reflecting the point $(2, 6)$ across the y-axis results in the new point $(-2, 6)$.
Folding a piece of paper in half and cutting a heart shape creates a symmetric figure where one half is a reflection of the other.

This is just like looking in a mirror! A reflection flips a shape over a line to create a perfect, reversed copy. The new shape has the same size and form, but it’s facing the opposite way, just like your reflection is a mirror image of you. Every point is the same distance from the mirror line, but on the other side.