Proving Theorems about Triangles
The Lesson Activities will help you meet these educational goals:
· Content Knowledge—You will prove theorems about triangles.
· Mathematical Practices—You will make sense of problems and solve them and reason abstractly and quantitatively.
· 21st Century Skills—You will use critical-thinking and problem-solving skills.
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. Interior Angles of Triangles
You will use the GeoGebra geometry tool to study the interior angles of triangles. Go to interior angles of triangles, and complete each step below. If you need help, follow these instructions for using GeoGebra.
a. Record the measures of the interior angles of ∆ABC (α, β, and γ), and find their sum.
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α / β / γ / Sum of Interior Anglesb. Move vertices A, B, and C to create a triangle of your choice. Record the measures of the interior angles α, β, and γ for three different triangles. Then find the sum of the angle measures for each triangle.
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α / β / γ / Sum of Interior Anglesc. How does the sum of the interior angles change as you move the vertices of ∆ABC to create a triangle of your choice?
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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. Base Angles of Isosceles Triangles
You will use GeoGebra to study the base angles of an isosceles triangle. Go to base angles of isosceles triangles, and complete each step below.
a. Find the lengths of and What is the relationship between the lengths? What kind of triangle is ΔABC?
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b. Find and compare the measures of the base angles, and . What do you notice about the angle measurements?
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c. Now move vertices A, B, and C of ∆ABC to create a triangle of your choice. (Note: the tool will allow you to create only certain kinds of triangles.) Does the relationship between the base angles change as the vertices change? If so, how?
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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. Connecting Triangle Midpoints
You will use GeoGebra to see what occurs when a midsegment joins two sides of a triangle. Go to connecting triangle midpoints, and complete each step below.
a. Find the midpoint of and label it D. Find the midpoint of and label it E. Record the coordinates of points D and E. When you’re through, draw
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b. Measure and record the slopes and lengths of and
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c. Describe the relationship between and using geometric terms.
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d. Change the positions of vertices A, B, and C to create a triangle of your choice. Does the relationship between and change when the vertices of ∆ABC are modified? Explain.
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e. How might you prove your observations about the slope of a midsegment in part d using algebra and x- and y-coordinates? Briefly outline an approach using what you know about midpoints and parallel lines. Use the figure you created in GeoGebra to guide you.
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f. How might you prove your observations from part d about the length of a midsegment using geometric theorems? Briefly outline an approach using what you know about midpoints, parallel lines, and congruent triangles. Use the figure you created in GeoGebra to guide you.
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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.
4. Concurrent Triangle Medians
You will use GeoGebra to study concurrence of medians of a triangle. Go to concurrent triangle medians, and complete each step below.
a. Using the existing midpoints of each side of the triangle, create the three medians. What do you observe about the way the medians intersect?
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b. Place a point at the point of intersection of the three medians. Now change the positions of vertices A, B, and C, and observe how the medians change. Is there any change in the way the medians intersect?
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c. How might you prove your observations in part b using algebra and x- and y-coordinates? Briefly outline an approach using what you know about a midpoint and the slope of a line.
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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