Transcript AMTNYS 2013 Understanding Congruence with Reflections

Association of Mathematics Teachers of NY State
Kristin A. Camenga
Houghton College
November 7, 2013
Slides available at
http://campus.houghton.edu/webs/employees/kcamenga/teachers.htm
…you will learn how Common Core defines
congruence.
…you will use transformations to prove
congruence in a variety of situations.
…you should grow in confidence in the
Common Core approach to congruence.
…you will be ready to find more connections
between transformations and what you already
teach.
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G-CO.B: Understand congruence in terms of
rigid motions
G-CO.7: Use the definition of congruence in
terms of rigid motions to show that two
triangles are congruent if and only if
corresponding pairs of sides and
corresponding pairs of angles are congruent.
G-CO.8: Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of rigid
motions.
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A rigid motion is a function of the plane that
preserves angles and distances.
All rigid motions are reflections, rotations,
translations or some composition of the
three.
A rigid motion is also called an isometry,
which means “same measure”
With your neighbors, discuss the following
problem:
Show:
A
B
1. 𝐴𝐶 ≅ 𝐵𝐷 using
rigid motions
C
D
2. ∠𝐴𝐶𝐷 ≅ ∠𝐵𝐷𝐶
What do students need to know to solve this
problem?
“Two geometric figures are defined to be
congruent if there is a sequence of rigid
motions that carries one onto the other.”
Based on the definition of congruence, work
with your neighbor to justify the following
statement:
Two triangles are congruent
if and only if
corresponding pairs of sides and
corresponding pairs of angles are congruent.
Given:
𝐴𝐵≅ 𝐴′ 𝐵′
𝐴𝐶 ≅ 𝐴′ 𝐶′ ∠𝐴 ≅
∠𝐴′
We need to show a
sequence of rigid motions
that will map Δ𝐴𝐵𝐶 to
Δ𝐴′ 𝐵′ 𝐶 ′ so all
corresponding sides and
angles coincide.
Given: 𝐴𝐵≅ 𝐴′ 𝐵′,
𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′
Translate
△ 𝐴𝐵𝐶 by vector
𝐴𝐴′ so that 𝐴
coincides with
𝐴’.
Given: 𝐴𝐵≅ 𝐴′ 𝐵′,
𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′
Rotate △ 𝐴𝐵𝐶 by
∠𝐶𝐴𝐶′ around A
so that ray 𝐴𝐶
coincides with
ray 𝐴′ 𝐶′.
Since 𝐴𝐵 ≅ 𝐴′ 𝐶′,
𝐶 coincides with
𝐶′.
Given: 𝐴𝐵≅ 𝐴′ 𝐵′,
𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′
Reflect △ 𝐴𝐵𝐶
over 𝐴𝐵.
Since ∠𝐴 ≅ ∠𝐴’
and the rays 𝐴𝐵
and 𝐴′ 𝐵′ coincide
and are on the
same side of the
angle, ∠𝐴
coincides with∠𝐴’.
Given: 𝐴𝐵≅ 𝐴′ 𝐵′,
𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′
Reflect △ 𝐴𝐵𝐶
over 𝐴𝐵.
Since the angles
coincide, the
other rays 𝐴𝐶 and
𝐴′ 𝐶′ coincide.
Since 𝐴𝐶 ≅ 𝐴′ 𝐶′, 𝐶
coincides with 𝐶′.
Given: 𝐴𝐵≅ 𝐴′ 𝐵′,
𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′
Since all sides
and angles
coincide, △
𝐴𝐵𝐶 ≅△ 𝐴′ 𝐵′ 𝐶 ′ .
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This proof guarantees that anytime we have
SAS, there is a sequence of rigid motions that
maps one of the triangles to the other, so
they are congruent.
Therefore, when using SAS, we do not need to
use a sequence of rigid motions to show
congruence.
With your neighbors, find a set of rigid motions
that will show that the following criteria are
enough to prove triangle congruence. Make
sure to explain how you know all the
corresponding parts coincide!
 ASA
 SSS
Given: ∠𝐴 ≅ ∠𝐴′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐶 ≅ ∠𝐶′
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Translate & rotate as with SAS to
align 𝐴 with 𝐴′ and 𝐶 with 𝐶′.
If 𝐵 and 𝐵’ are on different sides
of 𝐴𝐶, reflect △ 𝐴𝐵𝐶 over 𝐴𝐶.
◦ Since ∠𝐴 ≅ ∠𝐴’ and 𝐴𝐶 and 𝐴′ 𝐶′
coincide and are on the same side
of the angle, ∠𝐴 coincides with ∠𝐴’.
◦ Since the angles coincide, the other
rays 𝐴𝐵 and 𝐴′ 𝐵′ coincide.
◦ Similarly, since ∠𝐶 ≅ ∠𝐶’ and 𝐴𝐶and
𝐴′ 𝐶′ coincide, ∠𝐶 coincides with ∠𝐶’
and the other rays 𝐶𝐵 and
𝐶 ′ 𝐵′ coincide.
◦ Since the pairs of rays coincide,
their intersections 𝐵 and 𝐵′ coincide.
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Since all sides and angles
coincide, △ 𝐴𝐵𝐶 ≅△ 𝐴′𝐵′𝐶′.
What is harder about SSS?
No angles to align additional sides!
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What is harder about SSS?
No angles to align additional sides!
Draw BB ′ .
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What is harder about SSS?
No angles to align additional sides!
Then use SAS!
For each of the following theorems, which
transformations show the result?
(Look for symmetries!)
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The base angles of an isosceles triangle are
congruent.
Vertical angles are congruent.
A parallelogram has opposite sides and
angles congruent.
If a quadrilateral has diagonals that are
perpendicular bisectors of each other, then it
is a rhombus.
If two lines are parallel, a transversal creates
congruent alternate interior angles.
Reflect over
the angle
bisector 𝐴𝐷.
How does this
compare to the
standard proof
using SAS?
Rotation by 180◦
around the point
of intersection
Rotate 180◦ around the midpoint
of one of the diagonals.
𝐴𝐵 ≅ 𝐵𝐶
𝐴𝐵 ≅ 𝐴𝐷
𝐵𝐶 ≅ 𝐶𝐷
So all four
sides are
congruent.
Rotate about the
midpoint of the
transversal
For each of the following quadrilaterals,
describe the rotations and reflections that carry
it onto itself:
 Parallelogram
 Rhombus
 Rectangle
 Square
What connections do you notice?
Parallelogram
180○ rotation
Rhombus
180○ rotation, 2 lines
of symmetry
(diagonals)
Rectangle
180○ rotation, 2 lines
of symmetry (through
midpoints of sides)
Square
180○ rotation, 4
lines of symmetry
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Builds on students’ intuitive ideas so they can
participate in proof from the beginning.
Encourages visual and spatial thinking, helping
students consider the same ideas in multiple
ways.
Serves as a guide for students to remember
theorems and figure out problems.
Reinforces properties of transformations and
makes the geometry course more connected,
both within itself and to algebra.
Motivates changing perspective between pieceby-piece and global approaches. (MP.7)
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Ask students to look for symmetry regularly!
When introducing transformations, apply
them to common objects and ask what the
symmetry implies about the object.
Use transformations to organize information
and remember relationships.
Share another method of proof for a theorem
already in your curriculum.
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The ideas of symmetry and transformation
have application in algebra as well.
This can help students connect algebra and
geometry in a new way.
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Show mxn=nxm,
 Represent mxn as an array of
dots with m rows and n columns.
 Rotate the array by 90 degrees
and you have n rows and m
columns, or nxm dots.
 Rotation preserves length & area,
so these are the same number!
Translations and reflections of graphs
 Odd & even functions
 Circles: x2 + y2 = r2
 Unit circle trigonometry:
sin(π/2-x) = cos(x)
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Questions?
Kristin A. Camenga
[email protected]
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Wallace, Edward C., and West, Stephen F.,
Roads to Geometry: section on
transformational proof
Henderson, David W., and Taimina, Daina,
Experiencing Geometry
NYS Common Core Mathematics Curriculum,
Geometry: Module 1
These slides can be found at
http://campus.houghton.edu/webs/employees/kcamenga/teachers.htm