Similar and Congruent Triangles

Core Concept

Similarity vs. Congruence

Two triangles are congruent ($\cong$) when they have exactly the same shape and size — all corresponding sides and angles are equal. Two triangles are similar ($\sim$) when they have the same shape but potentially different sizes — all corresponding angles are equal and all corresponding sides are in the same ratio.

The scale factor $k$: If $\triangle ABC \sim \triangle DEF$ with scale factor $k$, then every length in $\triangle DEF$ is $k$ times the corresponding length in $\triangle ABC$. Crucially, all angle measures stay unchanged when you scale a triangle.

Angle-Angle (AA) Similarity

Because the angles of any triangle sum to $180°$, if two pairs of corresponding angles are equal, the third pair must be equal too. This is the most common similarity trigger on the SAT:

If $\angle A = \angle D$ and $\angle B = \angle E$, then $\triangle ABC \sim \triangle DEF$.

Congruence Criteria

You must know four criteria (memorize these — none are on the reference sheet):

AAA is NOT a congruence criterion — it only guarantees similarity. SS (two sides only) is NOT sufficient for either similarity or congruence.

Nested Triangles and the Parallel-Segment Rule

When a segment connects two sides of a triangle and is parallel to the third side, it creates a smaller triangle nested inside the original. Both triangles share the top angle, and the parallel lines create equal corresponding angles (from your prerequisite knowledge of parallel lines and transversals). By AA, the two triangles are similar.

If $D$ is on $\overline{AB}$, $E$ is on $\overline{AC}$, and $DE \parallel BC$, then:

SAT move: Compute $k = \dfrac{AD}{AB}$ (part over whole, not part over part), then multiply or divide any side by $k$.