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.
Sample answer:
α / β / γ / Sum of Interior Angles90° / 53.13° / 36.87° / 180°
b. 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.
Sample answer:
Answers will vary based on the coordinates selected for A, B, and C.
α / β / γ / Sum of Interior Angles75.96° / 50.91° / 53.13° / 180°
45° / 126.87° / 8.13° / 180°
26.57° / 135° / 18.43° / 180°
c. How does the sum of the interior angles change as you move the vertices of ∆ABC to create a triangle of your choice?
Sample answer:
The sum of the measures of the interior angles remains 180° no matter how the triangle changes.
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?
Sample answer:
The lengths of and are equal: they both measure 4.47 units. Since two sides of ΔABC are congruent, ΔABC is an isosceles triangle.
b. Find and compare the measures of the base angles, and . What do you notice about the angle measurements?
Sample answer:
m = m = 63.43°. The measurements for the two base angles are equal.
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?
Sample answer:
The relationship between the base angles does not change. The measures of the base angles of the isosceles triangle remain equal even if the vertices of the isosceles triangle change.
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
Sample answer:
The coordinates of the midpoints are D(1.5, 2.5) and E(3.5, 2.5).
b. Measure and record the slopes and lengths of and
Sample answer:
The slopes of and are both 0.
The length of is 4, and the length of is 2.
c. Describe the relationship between and using geometric terms.
Sample answer:
and are parallel to each other, and is half the length of
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.
Sample answer:
No, there is no change in the relationship. The slopes of and continue to be equal, and the length of remains half of the length of
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.
Sample answer:
Answers will vary, but should include some of these points:
I can use variables to represent the coordinates of the vertices for a general triangle, ∆ABC. Then I can calculate the midpoints of the sides in terms of the same variables, and calculate the slope of each midsegment. Showing that the expression for the slope of a midsegment is the same as the expression for the slope of the third side of the triangle proves that the two are parallel.
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.
Sample answer:
Answers will vary, but should include some of these points:
To prove the relationship about the length of a midsegment, I can use the properties of parallel lines and transversals. If I draw at least two midsegments, I see that they form congruent triangles within the original triangle, ∆ABC. I can prove the congruence of these triangles using triangle congruence criterion, and then use properties of congruent triangles and the Transitive Property to show that the length of the midsegment is half the length of the side it is parallel to.
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?
Sample answer:
The three medians intersect at a single point.
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?
Sample answer:
No, the medians continue to intersect at a single point.
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.
Sample answer:
Answers will vary, but should include some of these points:
I can use variables to represent the coordinates of the vertices for a general triangle, ∆ABC. Then I can calculate the midpoints of the sides in terms of those variables. Using the point-slope formula for the equation of a straight line, I can build the symbolic equations for the three medians, and I can solve for the point of intersection for two of the medians, and for example. Finally, I can prove the lines (i.e., medians) concurrent if the point I found also satisfies the equation of the line for
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