Grade 8 Unit 2.1: Congruence and Similarity
Unit Focus
Student will:

Explore congruence and similarity

Perform rigid transformations (translations, rotations, reflections)

Perform non-rigid transformations (dilations)

Explore similar triangles

Discover relationships between interior and exterior angles

Make conjectures about two parallel lines cut by a transversal
Approximate Number of Days: 25 days
Essential Questions
1.
2.
3.
4.
5.
6.
7.
How can the coordinate plane help me understand properties of reflections, rotations,
and translations?
What is the relationship between reflections, rotations, and translations?
What information is necessary before I can conclude that two figure s are congruent?
What happens to the sides and angles of a shape that undergoes a rigid transformation?
How can you determine if two shapes are congruent?
How can you determine if two shapes are similar?
Focus Content Standards
Fluency Standards
Understand congruence and similarity using physical models, transparencies or geometry
software. (Major Cluster Standards)
8.G.1 Verify experimentally the properties of rotations, reflections, and translations:
A. Lines are taken to lines, and line segments to line segments of the same length.
B. Angles are taken to angles of the same measure.
C. Parallel lines are taken to parallel lines.
Analyze and solve linear equations and pairs of simultaneous linear equations
*8.EE.7 Solve linear equations.
A. Give examples of linear equations in one variable with one solution infinitely many
solutions, or no solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until an equivalent
equation of the from x = a, a = a, or a =b results (where a and b are different
numbers).
B. Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting
like terms.
8.G.2 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; given two
congruent figures, describe a sequence that exhibits the congruence between them.
8.G.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional
figures using coordinates.
8.G.4 Understand that a two-dimensional figure is similar to another if the second can be
obtained from the first by a sequence of rotations, reflections, translations, and dilations; given
two similar two-dimensional figures, describe a sequence that exhibits the similarity between
them.
8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of
triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles. For example, arrange three copies of the same triangle
so that the sum of the three angles appears to form a line, and give an argument in terms of
transversals why this is so.
Unit Summative Assessment
Solve real-world mathematical problems involving volume of cylinders, cones and spheres.
**8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to
solve real-world and mathematical problems.
*Standard 8.EE.7 is first introduced in Unit 3.1. Students have been working informally with
one-variable linear equations since as early as kindergarten. This important line of
development culminates in grade 8, and students should have ample opportunities to develop
fluency with this standard by the end of Grade 8.
**Standard 8.G.9 is first introduced in Unit 1.1. When students learn to solve problems
involving volumes of cones, cylinders, and spheres - together with their previous grade 7 work
with angle measure, area, surface area and volume – they will have acquired a well-developed
set of geometry measurement skills. This standard should continue to be spiraled throughout
the Grade 8 year.
Standards for Mathematical Practice
Suggested Assessment (You may use this one or something different):
Note: These standards should drive your pedagogical practice every day.
“Company Logo” http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C16671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf - this task is aligned to HS Geometry
standards but can be modified to support the standards listed below.
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.
Design a Logo

Students explore common logos and describe any rigid transformations or dilations
observed in words. See below for some common logos:
o http://www.logoguru.co.uk/blog/famous-logos-with-simple-shapes/
o http://www.inspirationbit.com/8-bits-of-perfect-geometry-in-classic-logos/

Students use inspiration from these logos to create and describe a logo that involves a
rigid transformation and dilation and describe in detail the transformations performed
within the logo.

Students work to enlarge or shrink their logo for different products that the company
will use as promotional items.
Resources
DCPS Resources

McGraw Hill, Glencoe Math
o Chapter 6, Lessons 1, 2, 3, 4
o Chapter 7, Lessons 1, 2, 3, 4
Manipulatives

Nets of 3-dimensional shapes

3-dimensional solids

Geoboards

Graph paper

Tangrams

Ruler

Protractor

Centimeter Cubes

Inch cubes

Square tiles

Graphing calculators
Key Mathematical Vocabulary (Academic Language)
















Adjacent
Alternate interior/exterior angle
Angle
Angle sum
Angle-angle criterion
Clockwise/Counterclockwise
Congruent
Coordinate/Coordinate Plane/Grid
Corresponding Angles/Sides
Dilation
Image
Interior/Exterior Angle
Line/Line Segment
Parallel/Parallel Lines
Polygon
Reflection










Rotation
Scale factor/Factor
Similar/Similarity
Supplementary angles
Transformation
Translation
Transversal
Vertex
Vertical Angles
X-axis/Y-axis
Grade 8: Unit 2.1 Standards
Standard 8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
A. Lines are taken to lines, and line segments to line segments of the same length.
B. Angles are taken to angles of the same measure.
C. Parallel lines are taken to parallel lines.
Critical Knowledge and Subskills
Students will know that:

A rotation is a transformation that turns a figure around a
point

A reflection is a transformation that flips a figure across a line

A translation is a transformation that slides each point of a
figure the same distance in the same direction

Parallel lines are two lines on a plane that never meet. They
are always the same distance apart

Rotations, reflections, and translations are called rigid
transformations because they do not change the size or shape
of a figure

Multiple transformations can be performed on an image

When a transformation or series of transformations is
performed on a pre-image, the resulting image is congruent
to the pre-image
Students will be able to:

Perform a transformation/series of transformations on a
figure

Describe a transformation or series of transformations
performed on a shape

Identify corresponding angles from an image and a pre-image

Identify corresponding sides from an image and pre-image
Possible Teaching and Learning Tasks

Aaron’s Designs http://map.mathshell.org.uk/materials/tasks.php?taskid=361&subpage=apprentice – students
perform transformations of a shape on a coordinate plane.

“Patterns” http://academic.sun.ac.za/mathed/malati/Sec02.pdf (p11 -13) Students explore different patterns
made by through rigid transformations of a triangle.

Describe a sequence of transformations that makes Figure A congruent to Figure A’. (Borrowed from NC Unpacked
Standards.
Solution: Figure A’ was produced by a 90⁰ rotation clockwise around the origin.
Supplemental Resources
Digital:
http://www.sophia.org/translations-rotations-reflections-and-dilations-tutorial
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=1132&id=17249
http://www.geogebratube.org/material/show/id/124
Standard 8.G.2
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; given two congruent figures, describe a sequence that exhibits the congruence between them.
Critical Knowledge and Subskills
Students will know that:

Congruent figures are figures that are the exact same shape
and size

Two figures are congruent if one can be obtained from the
first by a sequence of rotations, reflections and translations
Possible Teaching and Learning Tasks

8.G Congruent Rectangles http://illustrativemathematics.org/illustrations/1228
Below is a picture of two congruent rectangles:
Students will be able to:

Prove two images are congruent by describing the sequence
of transformations that occurred from one image to the
other

Determine if two images are congruent by examining them
to see if a sequence of transformations can be made to get
from the pre-image to the image

Perform a sequence of transformations on a shape
a.
b.
c.

Show that the rectangles are congruent by finding a translation followed by a rotation which maps one of the
rectangles to the other.
Explain why the congruence of the two rectangles can not be shown by translating Rectangle 1 to Rectangle 2.
Can the congruence of the two rectangles be shown with a single reflection? Explain.
8.G Congruent Segments http://illustrativemathematics.org/illustrations/646
Line segments AB and CD have the same length. Describe a sequence of reflections that exhibits a congruence
between them.
Supplemental Resources
Digital:
http://virtualnerd.com/geometry/fundamentals/measuring-segments/congruent-definition
http://virtualnerd.com/geometry/congruent-triangles/congruence/triangle-corresponding-parts-example
Standard 8. G.3
Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
Critical Knowledge and Subskills
Students will know that:

Rigid transformations (translations, rotations, reflections) do
not change the shape or size of a figure. The image and preimage are congruent.

Dilation is a similarity transformation (non-rigid) where a
figure is enlarged (by a scale factor greater than one) or
reduced (by a scale factor less than one) without altering
the center

The result of a dilation is a similar image to the pre-image

Coordinates tell the location of different

Figures are rotated around a fixed point (often times the
origin)

Figures can be rotated clockwise or counterclockwise

Figures can be rotated a specific number of degrees (90⁰,
180⁰ and 270⁰ being the most common)

Figures are reflected over a line (the x-axis and y-axis are the
lines appropriate for 8th grade)
Possible Teaching and Learning Tasks

8.G Reflecting Reflections http://illustrativemathematics.org/illustrations/1243
Below is a picture of a triangle on a coordinate grid:
Students will be able to:

Given a pre-image, describe the effect and name the
resulting coordinates of the image that has been
transformed (dilated, translated, rotated, or reflected or a
combination of them).

Given an image and a transformation (or series of
transformations) describe and name the coordinates of the
pre-image

Given a pre-image and image, name the transformations
that occurred
1.
2.
3.

Draw the reflection of △ABC over the line x=−2. Label the image of A as A′, the image of B as B′ and the image
of C as C′.
Draw the reflection of △A′B′C′ over the line x=2. Label the image of A′ as A′′, the image of B′ as B′′ and the image
of C′ as C′′.
What single rigid transformation of the plane will map △ABC to △A′′B′′C′′? Explain.
Dilation and Similarity Task
Plot the ordered pairs given int eh table to make six different figures. Draw each figure on a separate sheet of graph
paper. Connect the points with line segments as follows:
o For Set 1, connect the points in order. Connect the last point in the set to the first point in the set.
o For Set 2, connect the points in order. Connect the last point in the set to the first point in the set.
o For Set 3, connect the points in order. Do not connect the last point in the set to the first point in the set.
o For Set 4, make a dot at each point (do not connect the dots)
After drawing the six figures, compare Figure 1 to each of the other figures and answer the following questions.
1.
2.
3.
4.
5.
6.
Which figures are similar? Explain your thinking.
Describe any similarities and/or differences between Figure 1 and each of the other figures.
a. Describe how corresponding sides compare.
b. Describe how corresponding angles compare.
How do the coordinates of each figure compare to the coordinates of Figure 1? If possible, write general rules for
making Figures 2-6.
Is having the same angle measures enough to make two figures similar? Why or why not?
What would be the effect of multiplying each of the coordinates in Figure 1 by ½?
Translate, reflect, rotate (between 0 and 90⁰), and dilate Figure 1 so that it lies entirely in Quadrant III on the
coordinate plane. You may perform the transformations in any order that you choose. Draw a picture of the new
figure at each step and explain the procedures you followed to get the new figure. Use coordinates to describe
the transformations and give the scale factor you used. Describe the similarities and differences between your
new figures and Figure 1.
Figure 1
Set 1
(6, 4)
(6, -4)
(-6, -4)
(-6, 4)
Set 2
(1, 1)
(1, -1)
(-1, -1)
(-1, 1)
Set 3
(4, -2)
(3, -3)
(-3, -3)
(-4, -2)
Figure 2
Set 1
(12, 8)
(12, -8)
(-12, -8)
(-12, 8)
Set 2
(2, 2)
(2, -2)
(-2, -2)
(-2, 2)
Set 3
(8, -4)
(6, -6)
(-6, -6)
(-8, -4)
Figure 3
Set 1
(18, 4)
(18, -4)
(-18, -4)
(-18, 4)
Set 2
(3, 1)
(3, -1)
(-3, -1)
(-3, 1)
Set 3
(12, -2)
(9, -3)
(-9, -3)
(-12, -2)
Figure 4
Set 1
(18, 12)
(18, -12)
(-18, -12)
(-18, 12)
Set 2
(3, 3)
(3, -3)
(-3, -3)
(-3, 3)
Set 3
(12, -6)
(9, -9)
(-9, -9)
(-12, -6)
Figure 5
Set 1
(6, 12)
(6, -12)
(-6, -12)
(-6, 12)
Set 2
(1, 3)
(1, -3)
(-1, -3)
(-1, 3)
Set 3
(4, -6)
(3, -9)
(-3, -9)
(-4, -6)
Figure 6
Set 1
(8, 6)
(8, -2)
(-4, -2)
(-4, 6)
Set 2
(3, 3)
(3, 1)
(1, 1)
(1, 3)
Set 3
(6, 0)
(5, -1)
(-1, -1)
(-2, 4)
Supplemental Resources
Digital:
http://www.gradeamathhelp.com/transformation-geometry.html
http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm
http://www.jmap.org/JMAP/RegentsExamsandQuestions/PDF/WorksheetsByPI-Topic/Geometry/Transformational_Geometry/G.G.58.Dilations.pdf
Standard 8.G.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations,
and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Critical Knowledge and Subskills
Possible Teaching and Learning Tasks
Students will know that:

Similar figures have angles with the same measure and sides
that are proportional

All congruent figures are similar

Not all similar figures are congruent

If a figure can be obtained from the first by a sequence of
rotations, reflections, translations and dilations they figures
are similar

A dilation enlarges (using a scale factor greater than on) or
reduces (using a scale factor less than one) an image

Similar figures are produced by dilations
Students will be able to:

Perform dilations (both reducing and enlarging) on a figure

Perform a transformation (rotation, reflection, translation,
dilation) or sequence of transformations on a figure and
name the coordinates of the resulting figure

Given two similar two-dimensional figures, describe a
sequence of transformations (rotations, reflections,
translations, dilations) that exhibits the similarity between
them

Given an image and a sequence of transformations, name
the coordinates of the pre-image

Is Figure A similar to Figure A’? Explain how you know. (From
http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/8th.pdf)
Solution: Dilated with a scale factor or ½ then reflected across the x axis, making Figures A and A’ similar.

Describe the sequence of transformations that results in the transformation of Figure A to Figure A’.
Solution: 90∘ clockwise rotation, translate 4 right and 2 up, dilation of ½. In this case the scale factor can be found by using
the horizontal distances on the triangle (image = 2 units; pre-image = 4 units)
Supplemental Resources
Digital:
http://learnzillion.com/lessons/1336-prove-two-figures-are-congruent-after-a-series-of-reflections-rotations-or-dilations
http://learnzillion.com/lessons/1357-prove-two-figures-are-similar-after-a-dilation
http://learnzillion.com/lessons/1398-describe-a-sequence-of-transformations
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=5053&id=13150
Standard 8.G.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles
appears to form a line, and give an argument in terms of transversals why this is so.
Critical Knowledge and Subskills
Possible Teaching and Learning Tasks
Students will know that:

The sum of interior angles of a triangle is 180⁰

The sum of exterior angles of a triangle is 360⁰

The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two interior angles

A transversal is a line that cuts across two or more parallel
lines

Many different angles are formed when parallel lines are cut
by transversals and there are many different relationships
among those lines

The angle-angle postulate for similar triangles states that if
two angles of one triangle are congruent to two angles of
another triangle, then the two triangles are congruent
Students will be able to:

Prove that the interior angles in a triangle sum to 180⁰

Determine the missing angle measure in a triangle when
given two of the angle measures

Prove that the sum of exterior angles of a triangle are equal
to 360⁰

Conjecture and discover relationships between the angles
that occur when two parallel lines are cut by a transversal

Find the measures of missing angles when two parallel lines
are cut by a transversal

Explore and determine the relationships between two similar
triangles

Determine if two triangles are similar using the angle-angle
criterion

Find the Missing Angle http://illustrativemathematics.org/illustrations/56
In the picture below, lines l and m are parallel. The measure of angle ∠PAX is 31∘, and the measure of
angle ∠PBY is 54∘. What is the measure of angle ∠ APB?

From http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/8th.pdf
o You are building a bench for a picnic table. The top of the bench will be parallel to the ground. If m∠1= 148⁰,
find m ∠2 and m ∠3. Explain your answer.
Solution: Angle 1 and angle 2 are alternate interior angles, giving angle 2 a measure of 148⁰. Angle 2 and
angle 3 are supplementary. Angle 3 will have a measure of 32⁰ so the m ∠2 and m ∠3 = 180⁰
o
In the figure below Line X is parallel to Line YZ. Prove that the sum of the angles of a triangle is 180⁰.
Solution: Angle a is 35⁰ because it alternates with the angel inside the triangle that measures 35⁰. Angle c is
80⁰ because it alternates with the angle inside the triangle that measures 80⁰. Because lines have a measure
of 180⁰, and angles a + b + c form a straight line, then angle b must be 65  180 – (35 +80) = 65. Therefore,
the sum of the angles of the triangle is 35⁰ + 65⁰ + 80⁰
Supplemental Resources
Digital:
http://learnzillion.com/lessons/1057-find-the-measure-of-an-angle-in-a-triangle-using-the-other-two-angles
http://learnzillion.com/lessons/1229-find-the-measurement-of-an-exterior-angle
http://learnzillion.com/lessons/1248-find-the-side-length-of-a-triangle-using-angleangle-criterion
http://learnzillion.com/lessons/1236-find-the-measurements-of-corresponding-angles
http://learnzillion.com/lessons/1237-find-the-measurements-of-vertical-and-adjacent-angles
http://learnzillion.com/lessons/1241-find-the-measurements-of-alternate-interior-and-alternate-exterior-angles