Rigid & Non-Rigid Transformations, Symmetry, and Triangle Congruence

Rigid Transformations (Review from Last Week)

• Definition: geometric actions that preserve both size and shape of every figure involved.
• Core properties unchanged under any rigid transformation:
  • Side length & overall dimensions
  • All internal & external angle measures
  • Parallelism (if two lines are parallel before, they remain parallel after)
• Properties that may change:
  • Orientation of individual angles (e.g.
  arrow of an angle might point a new way)
  • Global orientation of the shape (the entire figure can face a new direction)
• Typical rigid moves: translations, rotations, reflections, and any composition of these.

Non-Rigid Transformations (Focus on Dilations)

• Definition: transformations where size and/or shape may change.
• Class example required: dilation (uniform scaling in/out from a center).
• Properties preserved in a dilation:
  • Angle measures
  • Parallelism
  • Shape orientation (clockwise/counter-clockwise order of vertices)
• Properties not preserved in a dilation:
  • Side lengths
  • Overall shape size (area, perimeter)
• Rule of thumb: dilations = stretching or shrinking while keeping geometry proportionally similar.

Sequences / Compositions of Transformations

• Apply one transformation at a time and track which properties survive each step.
  • E.g. rotate → reflect → dilate → translate.
  • After every stage, re-evaluate side lengths, angles, parallelism, orientation.
• When you finish the chain, the cumulative outcome is simply the combination of the interim effects.

Symmetry Refresher

• Reflective (Mirror) Symmetry: a line splits the figure into two mirror images.
• Rotational Symmetry: rotating around some center by a fixed angle (<$360^{\circ}$) maps the figure onto itself.
• Both symmetries help spot hidden rigid transformations in figures.

Congruence – Core Concept for Today

• Two figures are congruent if they have exactly the same size and shape.
• Formal test: one figure can be mapped onto the other using only rigid transformations (no stretching or shrinking).
• Everyday example: identical rubber duck bath toys from the same manufacturer—one may simply be turned differently in the tub.

Shape-Specific Congruence Rules

• Line segments: congruent ⇔ equal lengths.
• Circles: congruent ⇔ equal radii; e.g. $r=6\Rightarrow A=\pi r^2 = 36\pi$ for both circles.
• Squares: congruent ⇔ all four corresponding sides equal.
• Rhombi: congruent ⇔ all four sides and each pair of corresponding angles equal.
• General Polygons: must match every corresponding side & angle ( often too exhaustive; shortcuts exist for triangles ).

Full Triangle Congruence (Brute-Force Definition)

• For two $\triangle$s, if all three side lengths and all three angle measures correspond, the triangles are congruent.
• Orientation is irrelevant—figures may be rotated, reflected, or flipped.

Why Shortcuts Exist

• Solving for six separate measures on each triangle is time-consuming and sometimes impossible (information missing).
• Mathematicians proved that certain smaller data sets alone guarantee the remaining parts match as well.

The Four Valid Triangle-Congruence Criteria

1. SSS (Side-Side-Side)
   • If all three pairs of corresponding sides are equal, $\triangle ABC \cong \triangle DEF$.
   • This automatically forces the three angles to coincide by rigid geometry.

2. SAS (Side-Angle-Side)
   • Two pairs of corresponding sides equal and the included angle (the angle between those sides) equal.
   • Sequence matters: the A must be nested between the two S’s.
   • Diagram notation: side marks (one dash, two dash) + matching angle arc.
   • Invalid siblings: SSA or ASS do not prove congruence.

3. ASA (Angle-Side-Angle)
   • Two angles match and the included side (the side between those two angles) matches.
   • Again, order critical: the S lies between the two A’s.

4. AAS (Angle-Angle-Side)
   • Two angles match and a side adjacent to one of those angles (but not between them) matches.
   • Equivalent phrasing: the non-included side next to either angle matches.

Invalid / Non-Sufficient Combos (Know to Reject)

• AAA: equal angles but unknown side scale ⇒ yields similar triangles, not necessarily congruent.
• SSA / ASS: two sides and a non-included angle may lead to 0, 1, or 2 different triangles ⇒ inconclusive.
• Any other random ordering (e.g. ASA but wrong side position) fails.

Visual/Example Highlights Mentioned in Class

• Example congruent triangles: sides $5,4,4$; angles $80^{\circ},50^{\circ},50^{\circ}$ even if one faces left and the other faces right—still congruent by SSS + ABA (entire set known).
• Instructor drew triangles showing:
  • SAS success (two matched sides + included angle)
  • How shifting the equal angle away from between the sides destroys SAS validity.
• Re-used numeric demos:
  • $30^{\circ}$, $20^{\circ}$ with side $6$ (ASA success)
  • $20^{\circ}$, $60^{\circ}$ with side $5$ adjacent—not between—(AAS success).

Practical Connections & Upcoming Work

• Congruence criteria are building blocks for geometric proofs (two-column, paragraph, or flow).
• Expect to justify why sides/angles correspond and cite one of the four shortcuts.
• Symmetry problems, coordinate-geometry tasks, and transformation challenges rely on these congruence tools.

Tips from the Instructor

• Order matters: memorize the positioning of the letters in SAS, ASA, AAS.
• When presented with multiple transformations, isolate each step first.
• Always look for the smallest data set that guarantees congruence—saves time on quizzes & proofs.
• Respond quickly in interactive sessions (thumbs-up / “yes”) so the lecture can keep moving!

Looking Ahead

• More hands-on practice in Desmos or Khan Academy.
• Writing full proofs will draw on today’s four criteria.
• Homework: attempt the two bolded congruence practice problems provided at the end of the slide deck.