Topics
9. Dilations & Similarity
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United States of America
Grade 8

Investigation: Angle angle similarity

Lesson
Practice
Overview
Activity
Reflection

Investigate informal arguments to establish facts about triangle similiarity.

Objectives
  • To understand what information is needed to prove triangle similarity.
  • To create informal arguments to show that triangles are similar.
Materials
  • None

Angle angle similarity

Vocabulary
When you are working through the activity, be sure to use the following vocabulary:
  • Transformation
  • Translation
  • Rotation
  • Dilation
  • Reflection
  • Congruent
  • Similar
Step 1
Use the applet below alongside the following procedure:
Use the slider "Red triangle size" to adjust the size of the triangle so that the two triangles are different sizes. You may also choose to drag one of the big vertices in either triangle to re-position the two triangles. Notice that there are two pairs of angles that are marked congruent in the two triangles.
Step 2
Now drag the bottom slider "Slide me slowly" far enough across that only one transformation of the triangle occurs. Describe the properties of this transformation with a partner.
Stop and answer the following questions with your partner:
Are the side lengths of the triangle preserved under the transformations? Are the angle measures in the triangle preserved under the transformations? Has the orientation of the triangle changed under the transformations?
Step 3
Continue to drag the "Slide me slowly" slider far enough across that a second transformation occurs. Describe the properties of this transformation with a partner.
Step 4
Drag the "Slide me slowly" slider completely across to the right side. Describe the properties of the transformation that occurred.

Use the following applet:

Loading interactive...
Investigate
Consider the following questions once you have completed the above procedure.
1.
Would it be possible to use the slider to transform the triangle on the right so that one of its vertices corresponds with a vertex of the triangle on the left - using only one transformation - regardless of where you initially had these two triangles initially positioned?
2.
Would it be possible to use the slider to transform the triangle on the right such that the angles marked with two lines will lie on top of one another, regardless of the position that the triangle started off in?
3.
What if the orientation of the triangle on the left was different than the orientation of the triangle on the left? Is there another type of transformation you could do first, that would allow steps 2 and 3 above to result in the congruent angles lying on top of one another?
4.
Would the transformation in step 4 have been possible regardless of the original size and position of the triangle on the right?

Answer the questions below after you have completed the activity.

Discussion
1.
Come up with an informal argument to describe why the triangles in the applet are similar shapes. Share your argument with a partner and discuss what is similar and different between your arguments.

Outcomes

8.G.A.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.A.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.

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