Question

explore
exploring congruence of parts of transformed
figures
you will investigate some conclusions you can make when you know that two figures are congruent.
a fold a sheet of paper in half. use a straightedge to draw a triangle on the folded sheet.
then cut out the triangle, cutting through both layers of paper to produce two congruent
triangles. label them △abc and △def, as shown.
b place the triangles next to each other on a desktop. since the triangles are congruent, there
must be a sequence of rigid motions that maps △abc to △def. describe the sequence of
rigid motions.
c the same sequence of rigid motions that maps △abc to △def maps parts of △abc to
parts of △def. complete the following.
\\( \overline { a b } \
ightarrow \\)
\\( \overline { b c } \
ightarrow \\)
\\( \overline { a c } \
ightarrow \\)
\\( a \
ightarrow \\)
\\( b \
ightarrow \\)
\\( c \
ightarrow \\)
d what does step c tell you about the corresponding parts of the two triangles? why?

Explanation:

Step1: Analyze the rigid motion for part B

Since the two triangles are congruent and cut from folded paper, a reflection (flip) over an appropriate line (the line of the fold) can map \( \triangle ABC\) to \( \triangle DEF\).

Step2: Determine the mappings for part C

For line - segments:

• If we consider the congruence mapping, by the property of congruent triangles (corresponding parts of congruent triangles are congruent), \(\overline{AB}\to\overline{DE}\), \(\overline{BC}\to\overline{EF}\), \(\overline{AC}\to\overline{DF}\)

For vertices: \(A\to D\), \(B\to E\), \(C\to F\)

Step3: Analyze the conclusion for part D

Step C tells us that the corresponding parts (sides and vertices) of the two congruent triangles are congruent. Because a rigid motion (in this case, a reflection) preserves distance (for sides) and position (for vertices). A rigid motion does not change the length of line - segments or the identity of points in terms of their geometric relationships.

Answer:

• Part B: A reflection over the line of the fold.
• Part C: \(\overline{AB}\to\overline{DE}\), \(\overline{BC}\to\overline{EF}\), \(\overline{AC}\to\overline{DF}\), \(A\to D\), \(B\to E\), \(C\to F\)
• Part D: The corresponding parts (sides and vertices) of the two congruent triangles are congruent. Because rigid motions preserve distance (for sides) and the identity of points (for vertices).