Geometry Unit 2.1: Congruent Triangles
Approximate Number of Days: 18 days
Unit Focus
Essential Questions
During this unit students will:
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Experiment with transformation in the plane

Develop an understanding of congruence in terms of rigid motion and prove
geometric theorems.
What is rigid motion and how does congruence relate to it?
How can we show that triangles are congruent?
How does SSS, SAS, and ASA congruence criteria follow from the rigid motion definition?
Focus Content Standards
Fluency Standards
Experiment with transformations in the place
G.CO.5 Given a geometric figure and a rotation, reflection, or translation (no dilations), draw
the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a
sequence of transformations that will carry a given figure onto another.

G-SRT.5 Fluency with the triangle congruence and similarity criteria will help students
throughout their investigations of triangles, quadrilaterals, circles, parallelism, and
trigonometric ratios. These criteria are necessary tools in many geometric modeling
tasks.
Understand congruence in terms of rigid motions
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.

G-GPE.4, 5, 7 Fluency with the use of coordinates to establish geometric results,
calculate length and angle, and use geometric representations as a modeling tool are
some of the most valuable tools in mathematics and related fields.

G-CO.12 Fluency with the use of construction tools, physical and computational,
helps students draft a model of a geometric phenomenon and can lead to conjectures
and proofs.
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.
Prove theorems involving similarity
G.SRT.5 Use congruence for triangles to solve problems and to prove relationships in
geometric figures.
Prove geometric theorems
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a
triangle sum to 180°; 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
Use coordinates to prove simple geometric theorems algebraically
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. (Include triangles
and quadrilaterals to extend the lines and angles concepts.)
Standards for Mathematical Practice
Note: These standards should drive your pedagogical practice every day. The underlined standards are critical for this unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Resources
Key Mathematical Vocabulary (Academic Language)
DCPS Resources

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname:
OCMG

OCMG: Chapter 3

Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname:
LBKSG

LBKSG: Chapter 4 :Congruent Triangles , Chapter 5: Properties of Triangles,
Chapter 7: Transformations , Chapter 8: Similar Triangles
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Geometers Sketchpad
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GEOGEBRA (free software)
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Manipulatives
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TI Nspire calculators
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Protractors
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Compasses
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Straight edges
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Mirrors
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Patty paper
Websites and/or Additional Resources
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http://www.khanacademy.org/#geometry
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http://www.mathopenref.com/worksheetlist.html
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http://www/geogebratube.org
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http://illuminations.nct,.org
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http://education.ti.com/calculators/downloads/US/Activities
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http://shodor.org
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http://mathwarehouse.com
Unit Assessment
Acute
Adjacent
Congruence
Congruent Figures
Corresponding Pairs of
Angles
Corresponding Pairs of Sides
Equiangular
Equilateral
If and Only If
Isosceles
Obtuse
Reflexive
Right Triangle
Rigid Motion
Scalene
Symmetric
Transitive
Vertex
Congruence Through Rigid Motion
1. Triangle XYZ is plotted on the grid below.
2.
Shawn drew figure ABCD. He plans to create figure A'B'C'D' by translating figure
ABCD 6 units down and 4 units to the right. On the coordinate plane shown, draw
and label Shawn's figure A'B'C'D'.
Answer _____________________________
Part A
On the grid, draw the image of triangle XYZ after a clockwise rotation of
180º about the origin. Label the new triangle X'Y'Z'.
Part B
On the lines below, explain how you determined the location of point Y'.
______________________________________________________________
______________________________________________________________
3.
Alexis started making a design by drawing figure ABCD. The next figure in her design is
the reflection of figure ABCD in the y-axis. On the coordinate plane below, draw the
reflection of figure ABCD. Label the image A'B'C'D'.
4. The table below shows the coordinates of triangle RST and the coordinates of R' in triangle
R'S'T'. Triangle R'S'T' is a dilation of triangle RST.
Part A
What are the coordinates of points S' and point T'?
S' = (________,________)
T' = (________,________)
Geometry Unit 2.1 Standards
Standard G.CO.5
Given a geometric figure and a rotation, reflection, or translation (no dilations), draw the transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure onto another.
Critical Knowledge and Subskills
Possible Teaching and Learning Tasks
Students will know that:

Figures can be transformed by performing rotations,
reflections, and translations.

Transformations can carry a given figure onto itself.
Bike Trail Task
http://musingmathematically.blogspot.ca/2012/07/bike-trail-task.html
Students will be able to:

Draw figures that have been transformed by a rotation,
reflection, or translation.
G-CO Showing a triangle congruence: the general case
https://www.illustrativemathematics.org/illustrations/1549
Suppose △ABC and △PQR are distinct, congruent triangles. Using the steps below, show that a congruence can always be
shown with one, two, or three rigid motions:
a. A translation taking A to P (if necessary).
b.
c.
A rotation taking B, or the image of B after translation, to Q (if necessary).
A reflection about ⃡𝑃𝑄 which takes C, or its image after (a) and (b), to R (if necessary).
G-CO Showing a triangle congruence: a particular case
https://www.illustrativemathematics.org/illustrations/1547
Triangles ABC and PQR pictured below are congruent:
a.
Show the congruence using rigid motions of the plane.
b.
c.
d.
e.
Can the congruence be shown with a single translation, rotation, or reflection? Explain.
Is it possible to show the congruence using only translations? Explain.
Is it possible to show the congruence using only rotations? Explain.
Is it possible to show the congruence using only reflections? Explain.
Supplementary Resources
Resources

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG

OCMG: Chapter 3, 3-1

Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG

LBKSG: Chapter 7

Geometers Sketchpad

GEOGEBRA (Free Software)
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TI Nspire Calculators
DIGITAL RESOURCES:
TI NSPIRE Activities

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=17267

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=16035

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13128
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http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13133
Geometers Sketchpad Activities
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http://dl.dropbox.com/u/69014296/parallelogram_trans_practice.gsp
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http://dl.dropbox.com/u/69014296/mirror_and_reflection_000.pdf
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http://dl.dropbox.com/u/69014296/ferris_wheel_directions.pdf
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http://dl.dropbox.com/u/69014296/Tesselations_by_Translation.pdf
Geogebra Activities
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http://www.geogebratube.org/search/results/uid/27b022b7556
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http://www.geogebratube.org/search/results/uid/14e72aef5f
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http://www.geogebratube.org/material/show/id/1595
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http://www.geogebratube.org/material/show/id/2644
Shodor Activities
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http://compute2.shodor.org/interactivate/standards/organization/objective/2316/
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http://compute2.shodor.org/interactivate/standards/organization/objective/2317/
Math Warehouse Activities
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http://www.mathwarehouse.com/transformations/index.php
Standard 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.
Critical Knowledge and Subskills
Students will know that:

Transformed figures can be described

Figures can be congruent
Possible Teaching and Learning Tasks
Isosceles Triangles
http://fivetriangles.blogspot.com/2012/04/isosceles-triangles.html
Illuminations (NCTM) tasks and lessons:
Students will be able to:
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Use geometric descriptions to transform figures.
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Predict the effect of a given rigid motion on a given
figure.
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Use the definition of congruence in terms of rigid
motions to decide if two figures are congruent.
1.
Calculation Nation
2.
Classifying Transformations
3.
Describing Reflections
4.
Describing Rotations
5.
Dihedral Figures
6.
Finding What Doesn’t Change
7.
Frieze Patterns
8.
Recognizing Transformations
9.
Reflect on This
10. Reflections Across Two Mirror Lines
11. Relating Rotations to Symmetry
12. Relationships Between Reflections and Symmetry
13. Shape Cutter
14. Symmetries I
15. Symmetries I: Conclusions
16. Symmetries II
17. Symmetries II: Conclusions
18. Symmetries III
19. Symmetries IV
20. Transformations and Frieze Patterns
Supplemental Resources
On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG
OCMG: Chapter 3, 3-1
Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 7: Transformations , Chapter 4: Congruent Triangles
Standard 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.
Critical Knowledge and Subskills
Students will know that:

Corresponding parts of congruent figures are in the same
position on each of figures.
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Congruent figures have the same shape and same size no
matter what the orientation of the two figures.
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Reflection, rotation and translation preserve congruence of
figures.
Possible Teaching and Learning Tasks
G-CO Properties of Congruent Triangles
https://www.illustrativemathematics.org/illustrations/1637
Below is a picture of two triangles:
Students will be able to:
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Identify corresponding sides and corresponding angles of two
congruent triangles.
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Determine whether a geometric figure has symmetry.
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Perform multiple transformations in a single problem.
a.
Suppose there is a sequence of rigid motions which maps △ABC to △DEF. Explain why corresponding sides and
b.
angles of these triangles are congruent.
Suppose instead that corresponding sides and angles of △ABC and DEF are congruent. Show that there is a
sequence of rigid motions which maps △ABC to △DEF.
Supplemental Resources
On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG
OCMG: Chapter 3, 3-2, 3-3
Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 7 Transformations, Chapter 4: Congruent triangles , Chapter 5: Properties of Triangles
DIGITAL RESOURCES:
TI NSPIRE Activities

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=8516

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=16882

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13158

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13160
Geometers Sketchpad Activities

http://sketchexchange.keypress.com/browse/topic/transformations-and-tessellations/by-recent/2/324/transformations
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http://sketchexchange.keypress.com/browse/topic/transformations-and-tessellations/by-recent/1/427/exploring-line-symmetry
Geogebra Activities
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http://www.geogebratube.org/material/show/id/7939
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http://www.geogebratube.org/material/show/id/9556
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http://www.geogebratube.org/material/show/id/2186

http://www.geogebratube.org/material/show/id/7897
Standard G.GPE.4
Use coordinates to prove simple geometric theorems algebraically. (Include triangles and quadrilaterals to extend the lines and angles concepts.)
Critical Knowledge and Subskills
Students will know that:

A rectangle is composed of two sets of parallel lines with
adjacent lines formed at right angles.
Students will be able to:

Prove whether a figure in the coordinate plane is a
rectangle.
Possible Teaching and Learning Tasks
G-GPE A Midpoint Miracle
http://www.illustrativemathematics.org/illustrations/605
Draw a quadrilateral ABCD. Try to draw your quadrilateral so that no two sides are congruent, no two angles are congruent,
and no two sides are parallel.
a.
b.
Let P, Q, R, and S be the midpoints of sides AB, BC, CD, and DA, respectively. Use a ruler to locate these points
as precisely as you can, and join them to form a new quadrilateral PQRS. What do you notice about the
quadrilateral PQRS?
Suppose your quadrilateral ABCD lies in the coordinate plane. Let (x1,y1) be the coordinates of
vertex A, (x2,y2) the coordinates of B, (x3,y3) the coordinates of C, and (x4,y4) the coordinates of D. Use
coordinates to prove the observation you made in part (a).
Supplemental Resources
On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG
OCMG: Chapter 3, 3-8, 3-9
Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 6: Quadrilaterals
http://www.dcps.learnzillion.com/lessons/285-prove-whether-a-figure-is-a-rectangle-in-the-coordinate-plane
http://www.dcps.learnzillion.com/lessons/286-prove-whether-a-point-is-on-a-circle
Standard G.SRT.5
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
Critical Knowledge and Subskills
Students will know that:

congruence and similarity criteria for triangles can be
used to prove relationships in geometric figures

congruence and similarity criteria for triangles can be
Possible Teaching and Learning Tasks
G-SRT Bank Shot
http://www.illustrativemathematics.org/illustrations/651
used to solve problems in geometric figures.
Pablo is practicing bank shots on a standard 4 ft.-by-8 ft. pool table that has a wall on each side, a pocket in each corner, and
a pocket at the midpoint of each eight-foot side.
Students will be able to:
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use congruence and similarity criteria for triangles to
solve for missing sides and angles in congruent triangles.
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use congruence and similarity criteria for triangles to
prove that triangles are congruent
Pablo places the cue ball one foot away from the south wall of the table and one foot away from the west wall, as shown in
the diagram below. He wants to bank the cue ball off of the east wall and into the pocket at the midpoint of the north wall.
a.
At what point should the cue ball hit the east wall?
b.
After Pablo practices banking the cue ball off of the east wall, he tries placing the eight-ball two feet from the east wall,
as shown in the diagram below, so that if he shoots the cue ball exactly as he did before, the cue ball will strike the
eight-ball directly and sink the eight-ball into the north pocket. How far from the north wall should Pablo place the
eight-ball?
Supplemental Resources
On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG
OCMG: Chapter 3, 3-4
Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 4: Congruent Triangles , Chapter 8: Similar Triangles
DIGITAL RESOURCES:
TI NSPIRE Activies

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13156

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13162

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13158

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13160
Geometers Sketchpad Activities

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/6/166/congruent-triangles-theorems

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/3/325/properties-of-isosceles-and-eqilateral-triangles

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/2/423/medians-and-centroid

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/3/319/angles-of-triangles
Geogebra Activities

http://www.geogebratube.org/material/show/id/1595

http://www.geogebratube.org/material/show/id/2644

http://www.geogebratube.org/material/show/id/7027

http://www.geogebratube.org/material/show/id/6070
Standard G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.
Critical Knowledge and Subskills
Possible Teaching and Learning Tasks
Students will know that:

There are steps in a formal proof that must be justified.
TV Space – 3Act Lesson
http://mrpiccmath.weebly.com/blog/3-acts-tv-space
Students will be able to:

Prove the measure of interior angles of a triangle have a
sum of 180°.

Prove the base angles of an isosceles triangle are
congruent.

Prove the segment joining midpoints of two sides of a
triangle is parallel to the third side and half the length of
the third side.

Prove the medians of a triangle meet at one point.
Proofs of Pythagorean Theorem (MARS Task)
http://map.mathshell.org/materials/lessons.php?taskid=419&subpage=concept
Paper Folding
http://fivetriangles.blogspot.com/2012/04/paper-folding.html
Supplemental Resources
On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG
OCMG: Chapter 3, 3-5 - 3-9
Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 5: Properties of Triangles
TI NSPIRE Activities

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13156

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13162

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13158

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13160
Geometers Sketchpad Activities

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/6/166/congruent-triangles-theorems

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/3/325/properties-of-isosceles-and-eqilateral-triangles

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/2/423/medians-and-centroid

http://sketchexchange.keypress.com/browse/topic/geometry-1/by-recent/3/319/angles-of-triangles
Geogebra Activities

http://www.geogebratube.org/material/show/id/1595

http://www.geogebratube.org/material/show/id/2644

http://www.geogebratube.org/material/show/id/7027

http://www.geogebratube.org/material/show/id/6070