Lesson 6-4: Understand Congruence

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 is a series of transformations applied one after another to a figure. When you combine multiple rigid motions (like a rotation followed by a translation), the final image always maintains perfect congruence with the original starting figure.

Examples

• Two Steps: Triangle $ABC$ is first rotated $90^\circ$ counterclockwise about the origin, then translated $3$ units right to create triangle $A''B''C''$.
• Three Steps: Pentagon $DEFGH$ is translated $2$ units up, then rotated $270^\circ$ clockwise about point $D'$, producing pentagon $D''E''F''G''H''$.

Explanation

When performing sequences, order matters, and you must apply each new move to the result of the previous move. To keep from getting lost, we use "prime notation" to track our steps. The original figure is plain ($A$). After step one, it gets a single prime ($A'$). After step two, it gets a double prime ($A''$). No matter how many prime marks a shape collects, if you only used rigid motions, the final shape is still $100\%$ congruent to the original.

Property

Two figures are not congruent (denoted $\not\cong$) if there is at least one pair of corresponding parts (sides or angles) that have unequal measures. If a side length $s_1 \neq s_2$ or an angle measure $\alpha_1 \neq \alpha_2$ for any matching pair, the figures are absolutely not congruent.

Examples

• Checking Sides: Consider $\triangle ABC$ with side $CA=7$ and $\triangle DEF$ with corresponding side $FD=8$. Since $CA \neq FD$ ($7 \neq 8$), we can immediately conclude that $\triangle ABC \not\cong \triangle DEF$.
• Checking Angles: A square has four $90^\circ$ angles, while a non-square rhombus might have angles of $80^\circ$ and $100^\circ$. Since their corresponding angles are not equal ($90^\circ \neq 80^\circ$), a square and a non-square rhombus are not congruent, even if their side lengths are identical.

Explanation

To prove that two figures are NOT congruent, you do not need to test every single transformation. You only need to find one single counterexample. If you find just one side that is longer, or just one angle that is wider, you can stop immediately. They are not congruent. Always make sure you are comparing apples to apples—for example, compare the longest side of one figure to the longest side of the other.