Lesson 3: Congruence

Property

Two figures are congruent if and only if they have the exact same size and the exact same shape. We write $\Delta ABC \cong \Delta DEF$ to show that triangle $ABC$ is congruent to triangle $DEF$.

Examples

• Two squares with side length 5 cm are congruent because they have identical size and shape.
• A triangle with sides 3 cm, 4 cm, and 5 cm is congruent to another triangle with the exact same side lengths, even if one is rotated or flipped.
• Two rectangles with dimensions 6 cm by 8 cm are congruent, regardless of their position or orientation on a page.

Explanation

Congruent figures are identical in every way except for their position in space. Think of congruent figures as exact clones of each other that can be moved around, flipped over, or turned without changing their inherent size or shape. When figures are congruent, every corresponding angle and side must be completely equal.

Property

A translation is a rigid motion, meaning it preserves the size and shape of a figure. The translated figure (the image) is always congruent to the original figure (the preimage).

If figure $F$ is translated to create image $F'$, then $F \cong F'$. This means all corresponding side lengths and angle measures are equal.

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.