Download Mod 1 - Aim #19 - Manhasset Public Schools

Aim #19: How do we define congruence and correspondence?
Do Now: Name the three rigid motions:
1) _____________
2) _______________
CC Geometry H
3) _________________
Rigid motions preserve _________________ and _______________________.
Congruence: Two figures in the plane are congruent if there exists a
finite composition of basic rigid motions that map one figure onto the other.
Examine ΔPQR being mapped onto ΔXYZ through a composition of rigid motions:
F
D
P'
U
P''
R'
Q'
E
Q''
R''
• To map ΔPQR to ΔXYZ, we first rotate ΔPQR 120 about point D (RD,120 ).
o
o
• Then reflect this image, ΔP'Q'R', across EF (rEF).
• Finally, translate ΔP"Q"R", along the given vector to obtain ΔXYZ (Tu).
Since each transformation is a rigid motion, ΔPQR ≅ ΔXYZ. We use function
notation to describe the composition of the rotation, reflection, and translation:
Tu (rEF (RD,120o (ΔPQR ))) = ΔXYZ
*Notice that the innermost function/transformation (the rotation) is performed
FIRST, and the outermost (the translation) LAST!
Every congruence gives rise to a correspondence.
A correspondence between two triangles is a pairing of each vertex of one triangle
with one and only one vertex of the other triangle. When reasoning about figures, it
is useful to be able to refer to corresponding parts (e.g., sides and angles) of the
two figures. We look at one part of the first figure and compare it to the
corresponding part of the other.
1. Name the corresponding angles and sides of the two triangles we see above:
corresponding angles
corresponding sides
2. Use your understanding of congruence to explain why a triangle cannot be
congruent to a quadrilateral.
A rigid motion F always produces a one-to-one correspondence between the points in
the pre-image and points in its image. If P is a point in the pre-image, then the
corresponding point in the image is F(P). A rigid motion maps each part of the
pre-image to a corresponding part of the image. As a result,corresponding
parts of congruent figures are congruent, since the very same rigid motion that makes
a congruence between the figures also makes a congruence between each part of the
figure and the corresponding part of the image.
*There are correspondences that do not come from congruences.
When we study similarity, the sides (and/or angles) of two figures might be
compared even when the figures are not congruent.
1) a. In the diagram below, the triangle on the left has been mapped to the one on the
right by a 240˚ rotation about P. Identify all six pairs of corresponding parts.
Corresponding
angles
Corresponding
sides
b. What rigid motion mapped ΔABC onto ΔXYZ? Write the transformation in
function notation.
2) ABCD is a square, and AC is one diagonal of the square. ΔABC is a reflection of
ΔADC across AC. Complete the table below:
a. Is ΔABC ≅ ΔADC? Justify your response.
b. Are the corresponding sides and angles congruent? Justify your response.
Each exercise below shows a sequence of rigid motions that map a pre-image onto a
final image. Identify each rigid motion in the sequence, writing the composition using
function notation. Write the congruence of each set of corresponding sides and
angles, proving that the pre-image is congruent to the final image by showing that
every side and every angle in the pre-image maps onto its corresponding side and
angle in the image. Finally, make a statement about the congruence of the pre-image
and final image.
C"
C'
3)
B
A
A"
B"
Sequence of Rigid Motions (2)
Composition in function notation
Sequence of corresponding sides
Sequence of corresponding angles
Triangle congruence statement
4)
Sequence of Rigid Motions (3)
Composition in function notation
Sequence of corresponding sides
Sequence of corresponding angles
Triangle congruence statement
B'
_____________
5)
Sequence of Rigid
Motions (2)
Composition in
function notation
Sequence of
corresponding
sides
Sequence of
corresponding
angles
Triangle congruence
statement
6) Give an example, using a diagram, of two quadrilaterals and a correspondence
between their vertices such that corresponding sides are congruent, BUT
corresponding angles are not congruent.
7) Assume that the following figures are drawn to scale. Use your understanding of
congruence to explain why square ABCD and rhombus GHIJ are not congruent.
Sum it Up!
• The phrase "same size same shape" may seem like a good enough definition for
figures being congruent, but it is not. When we use the term congruence, weneed
to refer to the rigid motion. We must show that there is a transformation that
maps one directly onto the other.
• A correspondence between 2 triangles is a pairing of each vertex of one triangle
with one and only one vertex of the other triangle. This pairing can also be
applied to figures other than triangles.
• A congruence naturally yields a correspondence since a rigid motion maps each
part of a figure to what we call a corresponding part of the image.
Name ____________________
Date ___________________
Z"
1.
CC Geometry H
HW #19
2.
C
A'
X"
Y"
B
Z
A"
Z'
B"
B'
A
X
X'
Y'
Y
B
A
Describe the sequence of rigid motions. Describe the sequence of rigid motions.
,
,
Composition :
(using notation)__________________
3.
4.
A'
B"
B
Composition :
(using notation)__________________
B'''
B
A"
A
C
C
A'
C'''
B"
A
Describe the sequence of rigid motions.
,
Composition :
(using notation)_______________________
Describe the sequence of rigid motions.
,
,
Composition :
(using notation)_______________________
5.
A'
Sequence of Rigid
Motions (3)
C'''
C
Composition in
function notation
B'''
B
Sequence of
corresponding sides
A
A'''
Sequence of corr. ≮s
Triangle congruence
statement
6. Give an example, using a diagram, of two triangles and a correspondence between
them such that all three corresponding angles are congruent BUT corresponding
sides are not congruent.
Review
1. Determine the correct name for:
a. ≮4 and ≮5 ___________________
b. ≮13 and ≮11 __________________
c. ≮8 and ≮6 ___________________
d. ≮15 and ≮11___________________
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>
2. Using a compass and straightedge, construct a line parallel to line l that goes
o
through point P using a rotation of 180
l
P