Lines, Angles, and Mathematical Proof

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will learn to write mathematical proofs and apply that knowledge to simple geometric relationships.
  • Mathematical Practices—You will reason abstractly and quantitatively and use appropriate tools strategically.
  • Inquiry—You will perform an investigationin which you will make observations and analyze results.
  • 21stCentury Skills—You will assess and validate information.

Directions

You will evaluatesome 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-CheckedActivities

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. Linear Pair and Supplementary Angles

You will use the GeoGebra geometry tool to explore the relationship between a linear pair and supplementary angles. Go to linear pairs andsupplementary angles, and complete each step below.If you need help, follow these instructions for using GeoGebra.

  1. and represent a linear pair because points A, D, and C lie on a straight line. Calculate the sum of and . Then move point B around and see how the angles change. What happens to the sum of and as you move point B around?

Type your response here:

  1. What can you say about and based on the sum of their angles? How are the angles related?

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  1. Based on your observations, what is the relationship between a linear pair and supplementary angles?

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How did you do? Check a box below.

Nailed It!—Iincludedall 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.

  1. Common Segments

You will use GeoGebra to empirically derive the Common Segments Theorem. Go to common segments, and complete each step below.

  1. Write down the lengths of and

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  1. Is there a relationship between the lengths of and ? If so, what is the relationship?

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  1. Now select point A, B, or D on the line segment and move the point while keeping the order of A, B, C, and D the same. How does the relationship between the lengths of the line segments change?

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  1. In general, what can you conclude about the lengths of and if the lengths of and are equal?

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How did you do? Check a box below.

Nailed It!—Iincludedall 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.

  1. Congruent and Supplementary Angles

You will use GeoGebra to empirically derive a theorem based on congruent and supplementary angles. Go to congruent and supplementary angles, and complete each step below.

  1. Study the measurements of and and list them below. If you move a point by mistake, press Ctrl + Z to revert back to the original setup.

Type your response here:

  1. What is the relationship between and Express your answer in geometric terms.

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  1. Now select point C and move it around. Do you observe any change in the relationship between the three angles?

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  1. Move point C so that and form a pair of supplementary angles, or angles whose sum is 180°. Record each of the angle measurements for this case.

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  1. In general, what can you conclude about a pair of angles that are both congruent and supplementary?

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How did you do? Check a box below.

Nailed It!—Iincludedall 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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