ASA, SAS, and SSS Criteria for Congruent Triangles

ASA, SAS, and SSS Criteria for Congruent Triangles

The Lesson Activities will help you meet these educational goals:

·  Content Knowledge—You will explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

·  Inquiry—You will perform an investigation in which you will make observations and draw conclusions.

·  21st Century Skills—You will assess and validate information.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1.  ASA Criterion

In this activity, you will use the GeoGebra geometry tool to create and compare triangles using the angle-side-angle criterion. Open GeoGebra, and complete each step below. If you need help, follow these instructions for using GeoGebra.

a.  Create a triangle of your choice on the grid. Measure two of the angles on the triangle and the length of the side between the two angles. Record the measurements in the table.

Type your response here:

Angle / Measure (°)
m
m
Side / Length

b.  Keep the triangle that you created in part a. You will now copy the triangle to a different location on the grid using two of its angles and the included side:

·  In a different place on the grid, draw a new line segment of the same length as the line segment recorded in part a. (This will be easier if you draw the new line segment parallel or perpendicular to the corresponding line segment from part a.)

·  At each endpoint of the new line segment, create an angle of a given size. Size the angles using the angle measurements you recorded in part a.

·  Once you’ve created the angles, draw a ray through each endpoint at the specified angle measurement.

·  Place a point at the intersection of the two rays.

·  Finish copying the triangle by drawing a polygon through the three points you created.

You should now have two triangles on the grid. What do you notice about the two triangles in terms of their size and shape?

Type your response here:

c.  Translations, rotations, and reflections are rigid transformations that preserve side lengths and angle measurements. Find and record a series of rigid transformations that will map the copied triangle to your original triangle. In the table, note the angles of rotation, lines of reflection, and distances moved. (Although the table provides space to record five steps, you might map your triangle in fewer steps.)

Type your response here:

Step / Transformation
1
2
3
4
5

d.  The triangle you created in part b should coincide with the original triangle you created. (If your triangles do not coincide, revisit the steps in parts a through c.) What does that mean in terms of congruency and rigid transformations? Explain. Paste a screen capture of your transformations in the space below.

Type your response here:

e.  Based on this activity, what is the minimum amount of information you need to define the shape and size of a triangle? How does this information establish congruency between two or more triangles?

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

2.  SAS Criterion

Open GeoGebra, and follow the steps below to create and compare triangles using the side-angle-side criterion.

a.  Create a triangle of your choice on the grid. Measure two of the sides on the triangle and the angle between the two sides. Record the measurements in the table.

Type your response here:

Side / Length
Angle / Measure (°)
m

b.  Keep the triangle that you created in part a. You will now copy the triangle to a different location on the grid using two of its sides and the included angle:

·  In a different place on the grid, draw a new line segment of the same length as one of those recorded in part a. (This will be easier if you draw the new line segment parallel or perpendicular to the corresponding line segment in the original triangle.)

·  At one of the endpoints of the new line segment, create an angle using the angle measurement that you recorded in part a.

·  Once you’ve created the angle, draw a ray through it.

·  Identify the other line segment you recorded in part a. On the ray, make a line segment equal in length to the identified segment. One way to do this is to draw a circle with a radius equal to the segment length. Then place a point at the intersection of the ray and the circle.

·  Finish copying the triangle by drawing a polygon through the three points you created.

You should now have two triangles on the grid. What do you notice about the two triangles in terms of their size and shape?

Type your response here:

c.  Find and record a series of rigid transformations that will map your copied triangle to your original triangle. In the table, note the angles of rotation, lines of reflection, and distances moved. (Although the table provides space to record five steps, you might map your triangle in fewer steps.)

Type your response here:

Step / Transformation
1
2
3
4
5

d.  The copy of your triangle should coincide with your original triangle. (If your triangles do not coincide, revisit the steps in parts a through c.) What does that mean in terms of congruency and rigid transformations? Explain. Paste a screen capture of your transformations in the space below.

Type your response here:

e.  Based on this activity, what is the minimum amount of information you need to define the shape and size of a triangle? How does this information establish congruency between two or more triangles?

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

3.  SSS Criterion

Open GeoGebra, and follow the steps below to create and compare triangles using the side-side-side criterion.

a.  Create a triangle of your choice on the grid. Measure the lengths of all the sides on the triangle and record them in the table.

Type your response here:

Side / Length

b.  Keep the triangle you created in part a. You will now copy the triangle to a different location on the grid using only its sides:

·  In a different place on the grid, draw a new line segment of the same length as one of those you recorded in part a. (This will be easier if you draw the new line segment in the same orientation as its corresponding line segment from part a.)

·  Next, copy the other two line segments from your original triangle. A reliable way to do this is to draw a circle centered on each endpoint of your new line segment. Make the radius of each circle equal to each of the lengths of the two remaining line segments on the original triangle. One of the intersection points of the two circles marks the third vertex of your new triangle. Place a point on the intersection.

·  Finish copying the triangle by drawing a polygon through the three points that you’ve established.

You should now have two triangles on the grid. What do you notice about the two triangles in terms of their size and shape?

Type your response here:

c.  Find and record a series of rigid transformations that will map your copied triangle to your original triangle. In the table, note the angles of rotation, lines of reflection, and distances moved. (Although the table provides space to record five steps, you might map your triangle in fewer steps.)

Type your response here:

Step / Transformation
1
2
3
4
5

d.  The copy of your triangle should coincide with your original triangle. (If your triangles do not coincide, revisit the steps in parts a through c.) What does that mean in terms of congruency and rigid transformations? Explain. Paste a screen capture of your transformations in the space below.

Type your response here:

e.  Based on this activity, what is the minimum amount of information you need to define the shape and size of a triangle? How does this information establish congruency between two or more triangles?

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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