Parallelogram

Both pairs of opposite sides are parallel.

**Fig. 1: ** Parallelogram

Fig. 1: Parallelogram

  • \(AB\parallel DC\)

  • \(CB\parallel DA\)

Rhombus

Parallelogram whose sides are all the same length.

**Fig. 2: ** Rhombus

Fig. 2: Rhombus

  • \(\overline{AB}=\overline{BC}=\overline{CD}=\overline{DA}\)

Isometry

Rotations, reflections, translations, and compositions of these that preserve distance, length, and angle measure. “rigid motion”

Translations

Move all the points across a vector \(\vec{v}\).

Notation: \(\tau_{\text{start point,end point}}\)

**Fig. 3: ** Translation of line AB along vector v.

Fig. 3: Translation of line AB along vector v.

Rotations

Pick a center O (origin), \(\theta\) (angle), takes P to P’ on a circle on a circle with center O and radius \(\overline{OP}\) with \(\angle POP'=\theta\).

Notation: \(R_{O,\theta}\)

**Fig. 4: ** Rotation of line OP.

Fig. 4: Rotation of line OP.

Reflections

Mirror of a shape across a line.

Pick line b.

  • Points on B don’t move.

  • Points not on b, P, go to P’ where b is perpendicular to bisector of \(\overline{PP'}\).

  • Midpoint m of PP’ on b make right angle \(\overline{PP'}\)

Notation: \(r_{b}\)

**Fig. 5: ** Triangle QPR reflected across line b.

Fig. 5: Triangle QPR reflected across line b.

Composition

Combinations of rotations, reflections, and translations.

Symmetry

An isometry that sends a geometric figure to itself.

Example:

Group

The set of isometries with composition is a group.

  1. Closure

    • Order doesn’t matter

    • \(ab=ba\)

  2. Associativity

    • Parentheses don’t matter

    • \((ab)c=a(bc)\)

  3. Identity

    • Anything combined with the identity equals itself.

    • \(ea=ae=a\)

  4. Inverses

    • undoes isometry

    • \(f^{-1}(f(a))=a\)