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. Congruent Triangles
In this activity, you will use the GeoGebra geometry tool to explore the relationships among the sides and angles of congruent triangles. If you need help, follow these instructions for using GeoGebra.

a. Transform ABC in these ways:
• Translate ABC by moving it 5 units to the right and 2 units up.
• Draw the line y = -x, and reflect ABC across the line.
• Rotate ABC counterclockwise about the origin by 270°.

When you’re through, capture the figure containing the preimage and the three images and paste it in the space provided.

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b. Measure the angles of ABC and its three images, and record the measurements in the table. Round each answer to the nearest degree.

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ABC
Measure A'B'C'
Measure A'1B'1C'1
Measure A'2B'2C'2
Measure
ABC
A'B'C'
A'1B'1C'1
A'2B'2C'2

ACB
A'C'B'
A'1C'1B'1
A'2C'2B'2

CAB
C'A'B'
C'1A'1B'1
C'2A'2B'2

c. Measure the lengths of the sides of ABC and its three images, and record the measurements in the table.

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ABC
Measure A'B'C'
Measure A'1B'1C'1
Measure A'2B'2C'2
Measure
AB A'B' A'1B'1 A'2B'2
BC B'C' B'1C'1 B'2C'2
AC A'C' A'1C'1 A'2C'2

d. How do the measures of the sides and angles of ABC compare with the corresponding measurements in the three images?

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e. Translations, rotations, and reflections are rigid transformations. What can you conclude about the measures of sides and angles on any triangle after undergoing a series of rigid transformations? Explain.

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f. By definition, two shapes are congruent if you can map one onto the other using rigid transformations (a sequence of one or more rotations, translations, and reflections). Since any sequence of rigid transformations performed on a triangle results in a congruent triangle, what does that imply about the corresponding side lengths and angle measures for two congruent triangles?

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g. If two triangles have three corresponding angles and three corresponding sides that are equal in measure, are the two triangles necessarily congruent? Why?

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