Everyone cut out a triangle ABC. It should be an asymmetrical triangle for best effect. Then whenever two congruent triangles were to be drawn, one could simply place the cardboard triangle on the paper and trace around it.
(This illustrates the first fundamental theorem of isometries – see reading in Brown, Theorem 10)
(1) Construction Problem: Given congruent triangles ABC and A'B'C', and any point D. Construct the point D' so that for the isometry T that takes ABC to A'B'C', the image T(D) = D'.
Method: Mark of distances DA, DB, DC with compass, then draw circles: circle with center A' and radius DA, circle with center B' and radius DB, circle with center C' and radius DC. The circles all pass through D'. See Brown, p. 42.
(This is the construction associated with the second fundamental theorem for isometries. Brown, Theorem 11.)
(2) Construction Problem: Given congruent triangles ABC and A'B'C', construct three (or fewer) lines so that the composition line reflections in the lines is an isometry that carries ABC to A'B'C'.
Method: This is explained in Brown, page 43. Each line is the perpendicular bisector of a certain point and the desired image of that point. The key fact that makes this work is that after the first step, we can assume that A coincides with A'. Then the perpendicular bisector of BB' goes through A by the locus property for perpendicular bisectors.
Definition of a glide reflection G -- invariant line and glide vector
Theorem: For every glide reflection G and any P, the midpoint of P and G(P) is on the invariant line of G.
Proof: This is our old theorem that the perpendicular bisector of a leg of a right triangle intersects the hypotenuse at the midpoint.
Consequence: Often an isometry is defined in other way than the definition (e.g., the composition of a rotation and a line reflection) but turns out to be a glide reflection. In this case, the invariant line and the glide vector can be discovered using this theorem.
(3) Experiment: Draw two congruent triangles ABC and A'B'C' by tracing around the cardboard triangle. Flip the triangle over after drawing ABC before drawing A'B'C'. Check that the midpoints of AA', BB', CC' are collinear. In this case there is a glide reflection taking ABC to A'B'C' and this is the invariant line (special case: if the invariant line is perpendicular to all three segments AA', BB', CC' then A'B'C' is a line reflection of ABC rather than a glide reflection).
Important Idea: Even through the glide reflection is typically defined as a composition of a translation and a line reflection, it is still one of the basic types of isometry, just like line translation or rotation. It is the isometry that you get if you pick up your triangle, flip if over and drop it down somewhere on the plane at random.
For lots of details about glide reflections, read Brown and the links on the Transformations Page.