Download Unit 8: Similarity, Congruence and Proofs

Accelerated GSE Algebra
1/Geometry A
TCSS Unit 8 Information
Curriculum Map: Similarity, Congruence, and Proofs
Content Descriptors:
Concept 1: Understand similarity in terms of similarity
transformations.
Concept 2: Prove theorems involving similarity.
Concept 3: Understand congruence in terms of rigid motions.
Concept 4: Prove Geometric Theorems.
Concept 5: Make geometric constructions.
Content from Frameworks: Similarity, Congruence, and Proofs
Unit Length: Approximately 26 days
20152016
TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
Curriculum Map
Unit Rational:
Building on standards from Coordinate Algebra and from middle school, students will use transformations and proportional reasoning to develop a
formal understanding of similarity and congruence. Students will identify criteria for similarity and congruence of triangles, develop facility with
geometric proofs (variety of formats), and use the concepts of similarity and congruence to prove theorems involving lines, angles, triangles, and
other polygons.
Prerequisites: As identified by the GSE Frameworks
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
Understand and use reflections, translations, and rotations.
Define the following terms: circle, bisector, perpendicular and parallel.
Solve multi-step equations.
Understand angle sum and exterior angle of triangles.
Know angles created when parallel lines are cut by a transversal.
Know facts about supplementary, complementary, vertical, and adjacent angles.
Solve problems involving scale drawings of geometric figures.
Draw geometric shapes with given conditions.
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the
first by a sequence of rotations, reflections, and translations.
Draw polygons in the coordinate plane given coordinates for the vertices.
Concept 1
Understand
similarity in terms of
similarity transformations.
GSE Standards
MGSE9-12.G.SRT.1 Verify
experimentally the properties of
dilations given by a center and a
scale factor:
a. A dilation takes a line not
passing through the center of the
dilation to a parallel line, and
leaves a line passing through the
center unchanged.
b. The dilation of a line segment is
longer or shorter in the ratio given
by the scale factor.
TCSS
Length of Unit
Concept 2
Prove theorems
involving similarity.
GSE Standards
MGSE9-12.G.SRT.4 Prove
theorems about triangles.
Theorems include: a line parallel
to one side of a triangle divides the
other two proportionally (and its
converse); the Pythagorean
Theorem using triangle similarity.
MGSE9-12.G.SRT.5 Use
congruence and similarity criteria
for triangles to solve problems and
to prove relationships in geometric
figures.
Concept 3
Understand
congruence in terms of
rigid motions.
GSE Standards
MGSE9-12.G.CO.6 Use geometric
descriptions of rigid motions to
transform figures and to predict the
effect of a given rigid motion on a
given figure; given two figures, use
the definition of congruence in terms
of rigid motions to decide if they are
congruent.
MGSE9-12.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.
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26 Days
Concept 4
Prove geometric
Theorems.
GSE Standards
MGSE9-12.G.CO.9 Prove
theorems about lines and
angles. Theorems include:
vertical angles are congruent;
when a transversal crosses
parallel lines, alternate
interior angles are congruent
and corresponding angles are
congruent; points on a
perpendicular bisector of a
line segment are exactly
those equidistant from the
segment’s endpoints.
Concept 5
Make
geometric
constructions.
GSE Standards
MGSE9-12.G.CO.12 Make
formal geometric
constructions with a variety
of tools and methods
(compass and straightedge,
string, reflective devices,
paper folding, dynamic
geometric software, etc.).
Copying a segment; copying
an angle; bisecting a
segment; bisecting an angle;
constructing perpendicular
lines, including the
perpendicular bisector of a
line segment; and
2
TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
MGSE9-12.G.SRT.2 Given two
figures, use the definition of
similarity in terms of similarity
transformations to decide if they
are similar; explain using similarity
transformations the
meaning of similarity for triangles
as the equality of all corresponding
pairs of angles and the
proportionality of all
corresponding pairs of sides.
CC9-12.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. (Extend to include
HL and AAS)
MGSE9-12.G.SRT.3 Use the
properties of similarity
transformations to establish the AA
criterion for two triangles to be
similar.
MGSE9-12.G.CO.10 Prove
theorems about triangles.
Theorems include: measures
of interior angles of a
triangle sum to 180 degrees;
base angles of isosceles
triangles are congruent; the
segment joining midpoints of
two sides of a triangle is
parallel to the third side and
half the
length; the medians of a
triangle meet at a point.
constructing a line parallel to
a given line through a point
not on the line.
MGSE9-12.G.CO.13
Construct an equilateral
triangle, a square, and a
regular hexagon, each
inscribed in a circle.
MGSE9-12.G.CO.11 Prove
theorems about
parallelograms. Theorems
include: opposite sides are
congruent, opposite angles
are congruent, the diagonals
of a parallelogram bisect
each other, and conversely,
rectangles are parallelograms
with congruent diagonals.
MGSE9-12.G.GPE.4
Use coordinates to prove
simple geometric theorems
algebraically.
For example, prove or
disprove that a figure defined
by four given points in the
coordinate plane is a rectangle; prove or
(Focus on quadrilaterals,
right triangles, and circles.)
Lesson Essential Question
Lesson Essential Question
 What is a dilation and how
does this transformation
affect a figure in the
coordinate plane?
 What strategies can I use to
determine missing side
lengths and areas of similar
figures?
TCSS
Lesson Essential Question
 Under what conditions are
similar figures congruent?
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Lesson Essential
Question
 How do I know which
method to use to prove
two triangles congruent?
 How do I know which
method to use to prove
two triangles similar?
Lesson Essential
Question
 In what ways can I use
congruent triangles to
justify many geometric
constructions?
 How do I make
geometric
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TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
Vocabulary
Dilations
Center
Scale Factor
Parallel lines
Line Segments
Ratio
Similarity
Transformations
Corresponding angles
Corresponding sides
Proportionality
AA criterion
TCSS
Vocabulary
Adjacent Angles
Alternate Exterior Angles
Alternate Interior Angles
Angle
Bisector
Centroid
Circum center
Coincidental
Complementary Angles
Congruent
Congruent Figures
Corresponding Angles
Corresponding Sides
Dilation
Parallel
Pythagorean Theorem
Endpoints
Equiangular
Similarity
Vocabulary
Equilateral
Exterior Angle of a Polygon
In center
Intersecting Lines
Intersection
Line
Line Segment or Segment
Linear Pair
Measure of each Interior Angle of
a Regular n-gon:
Orthocenter
Parallel Lines
Perpendicular Lines
Plane
Point
Proportion
Ratio
Ray
Rigid motions
Transform
Corresponding Angles
Corresponding Sides
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 How do I prove
geometric theorems
involving lines, angles,
triangles, and
parallelograms?
Vocabulary
Reflection
Reflection Line
Regular Polygon
Remote Interior Angles of
a Triangle
Rotation
Same-Side Interior Angles
Same-Side Exterior
Angles
Scale Factor
Similar Figures
Skew Lines
Sum of the Measures of
the Interior Angles of a
Convex Polygon
Supplementary Angles
Transformation
Translation
Transversal
Vertical Angles
Alternate interior
Perpendicular bisector
Equidistant
Endpoints
Theorems:
Interior angle sum
Theorem
Base angles of Isosceles
Triangle Theorem
Segments of midpoints of
a triangle Theorem
Medians of a triangle
Theorem
Median Isosceles Triangle
Midpoints
constructions?
Vocabulary
Construction
Segments
Angles
Bisect
Perpendicular lines
Perpendicular bisectors
Parallel lines
Equilateral triangle
Regular hexagon
inscribed
4
TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
Sample Assessment Items
Concept 1
Sample Assessment Items
Concept 2
Sample Assessment Items
Concept 3
Sample Assessment Items
Concepts 4
Sample Assessment Items
Concepts 5
MGSE9-12.G.SRT.1
MGSE9-12.G.SRT.4
MGSE9-12.G.CO.6
MGSE9-12.G.CO.9
MGSE9-12.G.CO.12
In the coordinate plane
segment M’N’ is the result of
a dilation of segment MN by
a scale factor of ⅓. Which
point is the center of
dilation?
Justify the last two steps of
the proof.
Parallelogram FGHJ was
translated 3 units down to form
parallelogram F’G’H’J’.
Parallelogram F’G’H’J’ was
then rotated 90°
counterclockwise about point
G’ to obtain parallelogram
F”G”H”JJ”.
What can be concluded if
Which diagram below shows
a correct mathematical
construction using only a
compass and a straightedge
to bisect an angle?
Given:
&
Prove:
R
A. (1, 3)
B. (0, 0)
C. (– 5 , 0)
D. (– 4 , 1)
1 7?
a.
S
T
a. t  p
b. p  q
c. p  q
U
MGSE9-12.G.SRT.2
Which transformation results
in a figure that is similar to
the original figure but has a
greater perimeter?
a. a dilation of
scale factor of 0.25
b. a dilation of
scale factor of 0.5
c. a dilation of
scale factor of 1
d. a dilation of
scale factor of 2
1. Given
2.
2. Given
3.
4.
3.
4.
by a
a. Symmetric Property of ; SSS
by a
b. Reflexive Property of ; SAS
by a
c. Reflexive Property of ; SSS
by a
d. Symmetric Property of ; SAS
MGSE9-12.G.SRT.3
Which method would it be
possible to prove the
triangles below are similar (if
TCSS
1.
MGSE9-12.G.SRT.5
In the triangle below, what is
the approximate value of x?
b.
d. p  t
MGSE9-12.G.CO.10
Which statement is true about
parallelogram FGHJ and
parallelogram F’G’H’J’?
a. The figures are similar but
not congruent.
b. The figures are congruent
but not similar.
c. The figures are both similar
and congruent.
d. The figures are neither
similar nor congruent.
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Match each statement in
the proof with the correct
reason.
A. Definition of midpoint
B. SAS congruence criterion
C. Corresponding parts of
congruent triangles are
congruent.
D. SSS congruence criterion
E. Definition of
perpendicular bisector
F. Given
G. If a point is on the
bisector of an angle, it is
equidistant from the sides
of the angle.
c.
d.
5
TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
MGSE9-12.G.CO.7
possible)?
For which pair of triangles
would you use AAS to
prove the congruence of
the 2 triangles?
MGSE9-12.G.CO.13
H. Reflexive property of
congruence
What is the first step when
inscribing a regular hexagon
in the circle below?
A)
a. SSS
b. AA
c. SAS
d Not similar
Given: AB  AC, M is the midpoint of BC
Prove: B  C
a.
b.
c.
d.
4 in
4.5 in
4.9 in
5.1 in
B)
____ 1. AB  AC
____ 2. BM  MC
____ 3. AM  AM
C)
____ 4.
ABM 
ACM
____ 5. B  C
D)
Answers: 1.F, 2. A, 3. H, 4.D, 5.C
MGSE9-12.G.CO.11
MGSE9-12.G.CO.8
Which statement is true about
ABC and FED?
a. Set the compass to any
distance. Then place the
point of the compass at A
and draw an arc that passes
through any point on the
circle.
In the accompanying
diagram of rhombus ABCD,
mCAB = 35°
What is
mCDA?
b. Place the point of the
compass at any point on the
circle and draw an arc that
passes through point A.
c. Use the compass to record
the radius of the circle.
d. Draw a segment from the
center of the circle to point
A.
a. The triangles are similar, but
not congruent.
b. The triangles are congruent
using ASA.
c. The triangles are congruent
using SAS.
TCSS
7/30/2015
A. 35⁰
B. 70⁰
C. 110⁰
D. 140⁰
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TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
d. The triangles are congruent
using SSS.
MGSE9-12.G.GPE.4
Given A (2,3), B (5, -1), C (1,
0), D (-4, -1), E (0,2), F (-1,-2)
Prove: ∠ABC≅∠DEF
Plot the points on a coordinate
plane.
Use the Distance Formula to find the
lengths of the sides of each triangle.
Resources – Concept 1
 Instructional
Strategies and
Common
Misconceptions
 Dilations practice
(G.SRT.1)
 Dilation drills G.SRT.1)
 Similar polygons
notes (G.SRT.2)
 Similar triangles
Graphic Organizer
(G.SRT.2)
TCSS
Key
Resources – Concept 2
Resources – Concept 3
 Instructional
Strategies and
Common
Misconceptions
 Congruence Graphic
Organizer (G.SRT.5)
 Property/Postulate/T
heorem “Cheat
Sheet”
 Properties matching
card game (activator)
 Triangle Bisector
notes (G.SRT.5)
 Instructional
Strategies and
Common
Misconceptions
 Congruency notes
(with proofs) (G.CO.7)
 Practice problems
with application
(G.CO.7)
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Resources – Concept 4

Instructional
Strategies and
Common
Misconceptions
 Properties of
Parallelograms notes
(G.CO.11)
Resources – Concept 5
 Instructional
Strategies and
Common
Misconceptions
 Constructing
lines/bisecting angles
notes and practice
 Proving properties of
(G.CO.12)
parallelograms
 How to bisect an
activity (G.CO.11)
angle (G.CO.12)
 Floor patterns
 How to copy an
extension
angle (G.CO.12)
activity(G.CO.9-11)
 How to construct a
grading rubric
line parallel to
7
TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
These tasks were taken
from the
GSE Frameworks.
These tasks were taken
from the
GSE Frameworks.
 Shadow Math –
extension activity
 Similar Triangles
(G.SRT.2,3,5)
Activator/Summarize
 Pythagorean
r (G.SRT.1,2,3)
Theorem using
Teacher
Student
Triangle Similarity
 Proving Similar
(G.SRT.4)
Triangles – guided
notes (G.SRT.3)
Teacher
Student
Textbook Resources
 Holt McDougal –
Explorations in Core
Math p 181184(G.SRT.1)
 Holt McDougal –
Explorations in Core
Math p 155-165
&168-170(G.SRT.2)
 Holt McDougal –
Explorations in Core
Math p 171176(G.SRT.3)
Textbook Resources
 Holt McDougal –
Explorations in Core
Math p 97-108, 177180 (G.SRT.4)
 Holt McDougal –
Explorations in Core
Math p
90,96,102(G.SRT.5)
These tasks were taken
from the
GSE Frameworks.
 Introducing
Congruence
activator/summarizer
(G.CO.6&7) Teacher
Student
 Proving Triangles
are Congruent station
activity (G.CO.6-8)
Teacher Student
Textbook Resources
 Holt McDougal –
Explorations in
Core Math p65-70
(G.CO.6)
 Holt McDougal –
Explorations in
Core Math p77-82
(G.CO.7)
 Holt McDougal –
Explorations in
Core Math p83,8696 (G.CO.8)
another line (G.CO.12)
These tasks were taken
 How to construct a
from the
perpendicular line
GSE Frameworks.
(G.CO.12)
 Lunch lines
notes/practice (G.CO.9)  Constructions
worksheet (G.CO.12)
Teacher
Student
 Centers of Triangles  Constructions
practice (G.CO.12)
(G.CO.10)
Teacher Student
 How to construct a
 Proving
hexagon and an
Quadrilaterals in the
equilateral triangle
Coordinate Plane
inscribed in a circle
(G.CO.11)
Teacher
Student
 Culminating Task:
Geometry Gardens –
project (extension
activity)
Textbook Resources
 Holt McDougal –
Explorations in Core
Math p35-54, 60-61,
84-85, 88 (G.CO.9) *not
numbers 1-3 on page 37 or 12 on page 38
(G.CO.13)
Textbook Resources
 Holt McDougal –
Explorations in Core
Math p5-16, 55-59,
62-64 (G.CO.12)
Holt McDougal –
Explorations in
Core Math p115120 (G.CO.13)
 Holt McDougal –
Explorations in Core
Math p122-130
(G.CO.10)
 Holt McDougal –
Explorations in Core
Math p131-154
(G.CO.11)
TCSS
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TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
Differentiated Activities
Concept 1
 Dilations in the
Coordinate Plane
(G.SRT.1&2)
Teacher
Student
Resources recommended
for Math Support
 Interactive
Vocabulary Site
(differentiate how vocabulary
is presented)
 What is similarity
(activator)
 G.SRT.1 GADOE
teacher notes
 G.SRT.2 GADOE
teacher notes
 G.SRT.3 GADOE
teacher notes
Differentiated Activities
Concept 2
Differentiated Activities
Concept 3
Differentiated Activities
Concept 4
Differentiated Activities
Concept 5
 Formalizing
Triangle
Congruence (G.CO.8)
Teacher
Student

Resources recommended
for Math Support
Resources recommended
for Math Support
Resources recommended
for Math Support
Resources recommended
for Math Support
Triangle Sum notes
 Analyzing
Congruence Proofs
FAL
 G.CO.6 GADOE
teacher notes
 G.CO.7 GADOE
teacher notes
These tasks were taken from
the
GSE Frameworks.
 G.CO.12 GADOE
teacher notes



Vocabulary/Postulate
Card match activity
Proof Scramble
These tasks were taken from
the
GSE Frameworks.

Triangle
Proportionality
Theorem (G.SRT.2-5)
Teacher
Student
Floodlights FAL application
(G.SRT.4&5, G.CO.9-11)
–
extension activity
 Evaluating
Statements about
length & area
FAL(G.CO.9-11)
These tasks were taken from
the
GSE Frameworks.
 Intro. activity
TCSS
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TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
At the end of TCSS Unit 5 student’s should be able to say “I can…” Take Away
 Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a segment
of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is
performed. However, a segment that passes through the center remains unchanged.
 Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the preimage, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor.
 Use the idea of dilation transformations to develop the definition of similarity.
 Given two figures determine whether they are similar and explain their similarity based on the equality of
corresponding angles and the proportionality of corresponding sides.
 Use the properties of similarity transformations to develop the criteria for proving similar triangles: AA.
 Use AA, SAS, SSS similarity theorems to prove triangles are similar.
 Prove a line parallel to one side of a triangle divides the other two proportionally, and its converse.
 Prove the Pythagorean Theorem using triangle similarity.
 Use similarity theorems to prove that two triangles are congruent.
 Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on
figures in the coordinate plane.
 Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea
of congruency and develop the definition of congruent.
 Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if their
corresponding sides and corresponding angles are congruent.
 Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria:
ASA, SSS, and SAS.
 Prove vertical angles are congruent.
 Prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are
congruent.
 Prove points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s
endpoints.
 Prove the measures of interior angles of a triangle have a sum of 180º.
 Prove base angles of isosceles triangles are congruent.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
 Prove the medians of a triangle meet at a point.
TCSS
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TCSS Unit 8 – Accelerated Algebra 1/ Geometry A
 Prove properties of parallelograms including: opposite sides are congruent, opposite angles are congruent, diagonals
of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
 Copy a segment and an angle.
 Bisect a segment and an angle.
 Construct perpendicular lines, including the perpendicular bisector of a line segment.
 Construct a line parallel to a given line through a point not on the line.
 Construct an equilateral triangle so that each vertex of the equilateral triangle is on the circle.
 Construct a square so that each vertex of the square is on the circle.
 Construct a regular hexagon so that each vertex of the regular hexagon is on the circle.
TCSS
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