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Geometry
Lesson Activities
AA, SAS, and SSS Criteria for Similar Triangles
The Lesson Activities will help you meet these educational goals:
 Content Knowledge— You will use the properties of similarity transformations to
establish the AA, SAS, and SSS criteria for two triangles to be similar.
 Mathematical Practices—You will use appropriate tools strategically.
 21st Century Skills—You will carry out technology-assisted modeling.
Directions
Please save this document before you begin working on the assignment. Type your
answers directly in the document.
_________________________________________________________________________
Teacher-Graded Activities
Write a response for each of the following activities. Check the Evaluation section at the end
of this document to make sure you have met the expected criteria for the assignment. When
you have finished, submit your work to your teacher.
1. AA Criterion for Similar Triangles, Part 1
You will use the GeoGebra geometry tool to investigate how many pairs of corresponding
angles in two triangles are enough to prove that the two triangles are similar. Open
GeoGebra, and complete each step below. If you need help, follow these instructions for
using GeoGebra.
a. Create a random triangle,  ABC. Measure and record its angles.
Type your response here:
Angle
Measure
 ABC
 ACB
 CAB
b. Now you will attempt to copy your original triangle using one of its angles:
 Draw a line segment, DE, of any length anywhere on the coordinate plane, but not
on top of  ABC.
 Choose one of the angles on  ABC. From point D, create an angle of the same
size as the angle you chose. Then draw a ray from D through the angle. You
should now have an angle that is congruent to the angle you chose on  ABC.
 Create a point anywhere outside the mouth, or opening, of the angle you created.
The point will initially be named F by the tool, but you should rename it point G.
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© 2013 EDMENTUM, INC.


Now draw a ray from E through G such that it intersects the first ray. Your creation
should be a closed shape resembling a triangle.
Label the point of intersection of the two rays F, and draw  DEF by creating a
polygon through points D, E, and F.
Click on point G, and move it around. By moving point G, you can change  DEF
and  EFD, while keeping  FDE fixed.
Take a screenshot of your results for one position of G, and paste it in the space
below.
Type your response here:
c. In GeoGebra, label the measures of the three angles on ∆DEF. Then move point G
around the coordinate plane to produce triangles with different sets of angles. Record
at least five sets of angles as you move G.
Type your response here:
Triangle m  DEF m  EFD
m  FDE
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2
3
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d. By moving point G, how many triangles is it possible to draw, keeping the measure of
just one angle constant (in this case, m  FDE)? In how many instances are all three
angle measures of  DEF equal to those of the original triangle,  ABC?
Type your response here:
e. To decide whether two triangles are similar, is it enough to know that one pair of
corresponding angle measures is equal? Use your observations and your
understanding of similarity transformations to explain your answer.
Type your response here:
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2. AA Criterion for Similar Triangles, Part 2
As you saw in the previous activity, knowing that one pair of corresponding angle
measures in two triangles is equal is not enough to prove that they are similar. Some
additional information must be required to prove triangle similarity.
Next, you’ll investigate whether two pairs of corresponding angles in two triangles having
equal angle measures is necessary and sufficient to prove that the triangles are similar.
Open GeoGebra again, and complete each step below.
a. Create a random triangle,  ABC. Measure and record its angles.
Type your response here:
Angle
Measure
 ABC
 ACB
 CAB
b. Now you’ll attempt to copy your original triangle using two of its angles:
 Choose two of the angles on  ABC, and locate the line segment between them.
Draw a new line segment, DE, parallel to the line segment you located on  ABC.
You can draw DE of any length and place it anywhere on the coordinate plane, but
not on top of  ABC.
 From points D and E, create an angle of the same size as the angles you chose on
 ABC. Then draw a ray from D and a ray from E through the angles such that the
rays intersect. You should now have two angles that are congruent to the angles
you chose on  ABC.
 Label the point of intersection of the two rays F, and draw  DEF by creating a
polygon through points D, E, and F.
Paste a screenshot of your results in the space below.
Type your response here:
c. Record all three angle measurements of  DEF in the table below.
Type your response here:
Angle
Measure
 DEF
 EFD
 FDE
d. How do the angles of  DEF compare with those in the original triangle? In particular,
compare the angle that you did not set in  DEF with the corresponding angle in
 ABC.
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Type your response here:
e. With two of the angles fixed on ∆DEF, what do you notice about the shape of ∆DEF
when compared with ∆ABC?
Type your response here:
f. Pick any side of  DEF, and find its length. Find the ratio (n) of this side to the
corresponding side of  ABC.
Type your response here:
g. Dilate  ABC about the origin using the scale factor n. Measure and compare the
lengths of the sides of the dilated triangle,  A′B′C′, with those of  DEF. Paste a
screenshot of your dilation below the table.
Type your response here:
Side
Length
Side
A'B '
DE
B'C'
EF
C ' A'
FD
Length
h. What can you conclude about  A′B′C′ and  DEF based on their side lengths, angle
measures, or both?
Type your response here:
i.
Explain why there must be a sequence of rigid transformations that will map  A′B′C′
exactly onto  DEF. Find and perform one such sequence of rigid transformations.
Describe the sequence of rigid transformations you performed.
Type your response here:
j.
Based on your responses to parts h and i, what can you conclude about  ABC and
 DEF? Explain your answer in terms of similarity transformations.
Type your response here:
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k. To decide whether two triangles are similar, is it enough to know that two pairs of
corresponding angle measures in the triangles are equal? Use your observations and
your understanding of similarity to explain your answer.
Type your response here:
3. SAS Criterion for Similar Triangles
The SAS criterion states that two triangles are similar if two pairs of corresponding sides
are in the same ratio and the angles included by the pairs of sides in the two triangles are
congruent. You’ll use GeoGebra to demonstrate this. Go to GeoGebra, and complete
each step below.
a. Create a random triangle,  ABC. Record the lengths of two of the sides of the triangle
and the measure of the included angle.
Type your response here:
b. In this step, you’ll attempt to copy your original triangle using only two of its sides and
the included angle. Follow these steps to construct the triangle:
 Draw a new line segment, DE, of any length, and place it anywhere on the
coordinate plane, but not on top of  ABC.
 Find and record the ratio of the length of this line segment to one of the
corresponding line segments on  ABC that you recorded in part a.
 Now multiply the ratio you calculated by the length of the other side of  ABC that
you selected in part a. Record the resulting length.
 Locate the endpoint on DE that corresponds with the vertex of the angle you
chose in part a. Using that point as the center, draw a circle with a radius equal to
the length you calculated in the previous step.
 From the center of the circle, draw a ray at an angle to DE . Make the angle
congruent to the angle of  ABC that you measured in part a.
 Mark the point of intersection of the ray and the circle, and label it point F.
 Complete ΔDEF by drawing a segment from F to the free endpoint of DE . Create a
polygon through points D, E, and F.
Paste a screenshot of your results below the measurements you record.
Type your response here:
c. Measure all the angles of ∆ABC and ∆DEF, and enter the measurements in the table.
Type your response here:
Angle
Measure
Angle
 CAB
 EDF
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Measure
 BCA
 FED
 ABC
 DFE
d. Does a relationship exist between the measures of the angles of ∆ABC and ∆DEF? If
so, explain how the triangles are related by these measurements. Use information that
you discovered earlier in these Lesson Activities.
Type your response here:
e. Your constructions for ∆ABC and ∆DEF were unique. Based on the random nature of
this activity, what conclusion can you draw about the similarity of two triangles when
two side lengths are proportional by the same ratio and the included angle is
congruent?
Type your response here:
4. SSS Criterion for Similar Triangles
The SSS criterion states that two triangles are similar if all three sides of one triangle are
proportional to the corresponding sides of the other triangle. You’ll use GeoGebra to
demonstrate this condition. Open GeoGebra again, and complete each step below.
a. Create a random triangle,  ABC, and record the lengths of its sides.
Type your response here:
Side
Length
AB
BC
CA
b. In this step, you’ll attempt to copy your original triangle using only its sides. Follow
these steps in the construction:
 Draw a new line segment, DE, of any length, and place it anywhere on the
coordinate plane, but not on top of ∆ABC.
 Find the ratio of the length of this segment to the length of one of the sides of
 ABC. Record the ratio.
 Multiply the ratio you calculated by the lengths of the other two sides of ∆ABC that
you selected in part a. Record the two resulting lengths.
 Create a circle centered at D with a radius equal to one of the lengths you
calculated in the previous step. Create a circle centered at E with a radius equal to
the other length you calculated in the previous step.
 Mark one of the points of intersection of the two circles, and label it point F. This
intersection marks the third point of the copied triangle. Complete ΔDEF by
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drawing line segments between points D, E, and F. Create a polygon through the
three points.
Paste a screenshot of your results below the measurements you record.
Type your response here:
c. Measure all the angles of ∆ABC and of ∆DEF, and enter the measurements in the
table.
Type your response here:
Angle
Measure
Angle
 ABC
 DFE
 BCA
 FED
 CAB
 EDF
Measure
d. Does a relationship exist between the measures of the angles of ∆ABC and ∆DEF? If
so, explain how the triangles are related through these measurements. Use
information that you discovered earlier in these Lesson Activities.
Type your response here:
e. Your constructions for ∆ABC and ∆DEF were unique. Based on the random nature of
this activity, what conclusion can you draw about the similarity of two triangles when all
three sets of corresponding sides of the triangles are proportional by the same ratio?
Type your response here:
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Evaluation
Your teacher will use these rubrics to evaluate the completeness of your work as well as the
clarity of thinking you exhibit.
Total Points: 100
1. Task 1: AA Criterion for Similar Triangles, Part 1
Task points: 15
a. Draw ∆ABC, and measure its angles.
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b. Draw ∆DEF using a segment and two rays.
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c. Record sets of angle measurements.
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d. Understand the limitations of one angle measurement.
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e. Make a conclusion about knowing only one angle measure.
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2. Task 2: AA Criterion for Similar Triangles, Part 2
Task points: 35
a. Draw ∆ABC, and measure its angles.
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b. Draw ∆DEF using a segment and two rays.
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c. Measure the angles of ∆DEF.
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d. Compare corresponding angles of ∆DEF and ∆ABC.
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e. Compare the shapes of ∆DEF and ∆ABC.
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f.
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Calculate the scale factor of the dilation, n.
g. Dilate ∆ABC, and record the lengths of corresponding sides.
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h. Make a conclusion about knowing two angle measures.
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i.
Transform dilated triangle ∆ABC, and map it to ∆DEF.
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j.
Make a conclusion about ∆ABC and ∆DEF.
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k. Make a conclusion about the AA criteria.
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3. Task 3: SAS Criteria for Similar Triangles
Task points: 25
a. Draw ∆ABC, and measure two sides and the included
angle.
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b. Construct ∆DEF.
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c. Record the angle measures of ∆ABC and ∆DEF.
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d. Recognize the relationship between corresponding angles
and its significance.
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e. Draw a conclusion about knowing two sides and the
included angle of a triangle.
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4. Task 4: SSS Criteria for Similar Triangles
Task points: 25
a. Draw ∆ABC, and measure its sides.
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b. Construct ∆DEF.
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c. Record the angle measures of ∆ABC and ∆DEF.
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d. Recognize the relationship between corresponding angles
and its significance.
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e. Draw a conclusion about knowing three sides of a triangle.
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