Similarity

GE.D.1Similar polygons01/04

Similar polygons — same shape, different size.

Step01of 04

Two polygons are similar (symbol $\sim$ ) if:

Corresponding angles	→are CONGRUENT
Corresponding sides	→are PROPORTIONAL

Both conditions must hold. Same shape, different size — like a photograph and an enlargement.

GE.D.1 — Similar polygons — same shape, different size.

Step 01 of 04

Two polygons are similar (symbol $\sim$ ) if:

Corresponding angles	→are CONGRUENT
Corresponding sides	→are PROPORTIONAL

Both conditions must hold. Same shape, different size — like a photograph and an enlargement.

Step 02 of 04

Scale factor. The ratio of corresponding sides — the same number for every pair.

If $△ A B C \sim △ D E F$ with $A B = 6, D E = 9$ , the scale factor from $A B C$ to $D E F$ is $9/6 = 1.5$ . Every other side of $D E F$ is $1.5 \times$ the corresponding side of $A B C$ .

Step 03 of 04

Worked example. Triangles $A B C$ and $D E F$ are similar with sides $A B = 4$ , $B C = 6$ , $C A = 8$ , and $D E = 6$ . Find $E F$ and $F D$ .

Scale factor: $6/4 = 1.5$ . So:

E F = 1.5 \cdot 6 = 9, F D = 1.5 \cdot 8 = 12

Step 04 of 04

Perimeter and area scale differently. If the linear scale factor is $k$ :

Side lengths	→scale by $k$
Perimeter	→scales by $k$
Area	→scales by $k^{2}$
Volume (3D)	→scales by $k^{3}$

A triangle scaled by 3× has 9× the area; a cube scaled by 2× has 8× the volume. This is a frequent test question.

Key insight

Similar = same angles, proportional sides. ONE scale factor governs every corresponding side. Areas grow by the SQUARE of the scale factor; volumes by the CUBE.

GE.D.2 — Similarity criteria — AA, SAS, SSS.

Step 01 of 04

You don't need to verify all 6 corresponding parts. Three criteria are enough.

AA (angle-angle)	→two pairs of corresponding angles congruent → similar
SAS (side-angle-side)	→two pairs of sides proportional AND included angles congruent → similar
SSS (side-side-side)	→all three pairs of sides proportional → similar

AA is the most common. If two angles match, the third must (angle-sum theorem), so all three angles are congruent — and the sides automatically come out proportional.

Step 02 of 04

Worked example — AA. Two right triangles share an acute angle measuring $35°$ . Both have a right angle. Therefore the triangles are similar by AA — both have $35°$ , both have $90°$ , so the third angle ( $55°$ ) is forced too.

Two right triangles sharing an acute angle — similar by AA.

Step 03 of 04

Why AA needs only TWO pairs but congruence (AAS) needs three. For congruence, AAA isn't enough — same angles can give different sizes. For SIMILARITY, that's exactly the point: angles only, sizes can differ.

Don't confuse the criteria families. SSS-congruence requires equal sides; SSS-similarity requires PROPORTIONAL sides.

Step 04 of 04

Set up the proportion correctly. Once you've shown two triangles similar, write the proportion with corresponding parts in the SAME order:

\frac{A B}{D E} = \frac{B C}{E F} = \frac{C A}{F D}

Cross-multiply to solve for an unknown side. Always pair-up matching vertices first; mismatched pairs lead to wrong answers even when the algebra is right.

Key insight

AA is enough for similarity (two angles force the third). SAS and SSS similarity use proportional sides, not equal ones — that's the only difference from the congruence versions. Always set up proportions in corresponding-vertex order.

GE.D.3 — Triangle proportionality — the side-splitter theorem.

Step 01 of 04

Side-splitter theorem. If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sidesproportionally.

DE ∥ BC splits AB and AC proportionally.

Step 02 of 04

The proportion.

\frac{A D}{D B} = \frac{A E}{E C}

Equivalently:

\frac{A D}{A B} = \frac{A E}{A C}

Both forms are correct — the first compares the two pieces to each other, the second compares each piece to the whole side. Pick whichever matches your given numbers.

Step 03 of 04

Worked example. In the figure above, $A D = 4, D B = 6, A E = 6$ . Find $E C$ .

\frac{A D}{D B} = \frac{A E}{E C} \Rightarrow \frac{4}{6} = \frac{6}{E C}

4 \cdot E C = 36 \Rightarrow E C = 9

Step 04 of 04

Why it's true. Triangles $A D E$ and $A B C$ share angle $A$ ; the parallel line forces corresponding angles to be congruent (because $D E ∥ B C$ ). So $△ A D E \sim △ A B C$ by AA, and proportional sides follow.

The converse also holds: if a line divides two sides of a triangle proportionally, that line must be parallel to the third side. Useful for proving lines parallel.

Key insight

Side splitter = "parallel line in a triangle gives proportional pieces." It's just AA similarity in disguise; recognizing the pattern saves a setup step. The converse lets you prove parallelism from a proportion.

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