8.3 Transformations, Congruence & Similarity Criteria

8.3 Transformations, Congruence & Similarity Criteria

Skills 9 and 10 of Competency 3 belong together: transformations define congruence and similarity, and the criteria are the shortcuts that let you establish them without exhibiting a transformation.

The four transformations

The first three are rigid motions (isometries): they preserve distance, so the image is congruent to the pre-image. A dilation preserves angles but not lengths, so the image is similar to the pre-image.

Coordinate rules

+---------------------------------------------------------------------------+
|  TRANSLATION    (x, y) -> (x + a, y + b)                                  |
|                                                                           |
|  REFLECTION     across x-axis:   (x, y) -> (x, -y)                        |
|                 across y-axis:   (x, y) -> (-x, y)                        |
|                 across y = x:    (x, y) -> (y, x)                         |
|                 across y = -x:   (x, y) -> (-y, -x)                       |
|                                                                           |
|  ROTATION about the origin, counterclockwise:                             |
|                 90 degrees:   (x, y) -> (-y, x)                           |
|                 180 degrees:  (x, y) -> (-x, -y)                          |
|                 270 degrees:  (x, y) -> (y, -x)                           |
|                                                                           |
|  DILATION centered at the origin, factor k:  (x, y) -> (kx, ky)           |
+---------------------------------------------------------------------------+

Two notes that items exploit. First, a 180° rotation and a point reflection through the origin produce the same result, (−x, −y), but a 180° rotation is not the same as reflecting across an axis. Second, a 270° counterclockwise rotation equals a 90° clockwise rotation, so read the direction carefully.

Reflect (3, −5) across the y-axis: (−3, −5). Rotate (3, −5) 90° counterclockwise about the origin: (5, 3). Dilate (3, −5) by k = 2 from the origin: (6, −10).

For a dilation, k > 1 enlarges, 0 < k < 1 reduces, and a negative k both scales and rotates 180°.

Congruence and similarity defined by transformations

Two figures are congruent if a sequence of rigid motions maps one exactly onto the other. Two figures are similar if a sequence of rigid motions and a dilation maps one exactly onto the other.

This is the modern definition, and items ask about it directly: "Which sequence of transformations shows that triangle ABC is similar to triangle DEF?" The answer must include a dilation whenever the sizes differ, and only rigid motions when they match.

Congruence implies similarity with scale factor k = 1. Similarity does not imply congruence.

Triangle congruence criteria

+---------------------------------------------------------------------------+
|  SSS   three pairs of congruent sides                                     |
|  SAS   two sides and the INCLUDED angle                                   |
|  ASA   two angles and the INCLUDED side                                   |
|  AAS   two angles and a NON-included side                                 |
|  HL    hypotenuse and one leg  -- RIGHT TRIANGLES ONLY                    |
+---------------------------------------------------------------------------+
|  NOT valid:                                                               |
|  SSA / ASS   the "ambiguous case" -- two different triangles possible     |
|  AAA         proves SIMILARITY only, not congruence                       |
+---------------------------------------------------------------------------+

"Included" is the whole distinction between SAS and SSA. In SAS the angle sits between the two sides. In SSA the angle is outside the pair, and that arrangement can produce two non-congruent triangles — the ambiguous case. Given sides 5 and 8 with a non-included 30° angle, the third vertex can land in two different places, giving two different triangles. This is why SSA is rejected.

HL is a special case that works because in a right triangle the hypotenuse and one leg determine the second leg via the Pythagorean theorem, effectively giving SSS. It applies only to right triangles.

AAA fails for congruence for an obvious reason: two equilateral triangles of different sizes have identical angles but different sizes. It is, however, sufficient for similarity — and in fact only two angles are needed.

Similarity criteria

+---------------------------------------------------------------------------+
|  AA    two pairs of congruent angles                                      |
|  SSS~  all three pairs of sides in the same ratio                         |
|  SAS~  two pairs of sides proportional with congruent INCLUDED angles     |
+---------------------------------------------------------------------------+

AA is sufficient because the Triangle Sum Theorem forces the third pair of angles to match as well.

Triangles with angles 50° and 70° in one and 50° and 60° in the other: the third angles are 60° and 70°, so all three pairs match and the triangles are similar by AA — even though the given pairs looked mismatched.

SSS similarity check: sides 6, 8, 10 and 9, 12, 15. Ratios 6/9 = 2/3, 8/12 = 2/3, 10/15 = 2/3. All equal, so the triangles are similar with scale factor 3/2.

Writing correspondence correctly

A congruence or similarity statement encodes the correspondence in the order of the letters. Writing △ABC ≅ △DEF asserts A ↔ D, B ↔ E, C ↔ F, so AB ≅ DE and ∠B ≅ ∠E. Items give a correct statement and ask which pair of parts must be congruent; matching positions in the two names is the entire technique.

Using similarity to find a missing length

△ABC ~ △DEF with AB = 8, BC = 12, and DE = 20. Find EF. Corresponding sides AB ↔ DE and BC ↔ EF, so 8/20 = 12/EF. Cross-multiply: 8 · EF = 240 → EF = 30.

The scale factor is 20/8 = 2.5, and indeed 12 · 2.5 = 30 — a useful independent check. Setting up the proportion with mismatched correspondence is the dominant error, which is why identifying the correspondence from the naming order comes first.