Proving Theorems about Parallelograms

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will prove theorems about parallelograms.
  • Inquiry—You will make observations, analyze results, 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

  1. Congruent Parts of Parallelograms

You will use GeoGebra to explore the side lengths and interior angles of parallelograms. Go to congruent parts of parallelograms, and complete each step below. If you need help, follow these instructions for using GeoGebra.

  1. Confirm that quadrilateral ABCD is a parallelogram. Measure and record the slope of each of its sides.

Sample answer:

Side / Slope (m)
/ 1.5
/ 0
/ 1.5
/ 0
  1. Measure and record the length of each side of parallelogram ABCD.

Sample answer:

Side / Length
/ 3.61
/ 2.0
/ 3.61
/ 2.0
  1. How are the lengths of the four sides of the parallelogram related?

Sample answer:

The opposite sides of the parallelogram have equal lengths.

  1. Measure and record the four interior angles of parallelogram ABCD.

Sample answer:

Angle / Measurement
BAD / 56.31°
ADC / 123.69°
BCD / 56.31°
ABC / 123.69°
  1. How are the angle measurements of the opposite interior angles related?

Sample answer:

The opposite interior angles of the parallelogram have equal measurements.

  1. Move vertices A, B, and C to modify the original parallelogram. As you change the parallelogram, notice what happens to the sides and the angles. How does moving the vertices affect the relationships between the sides and the angles that you noted in parts c and e?

Sample answer:

Despite changing the vertices, the opposite sides and opposite interior angles of the parallelogram remain equal.

  1. Explain how you can prove your observations in parts c and e using a traditional proof. You don’t need to develop the steps of the proof; simply describe an approach that you might use. Which theorems might appear in your proof? (Hint: Draw a diagonal on the parallelogram to assist you in describing an approach.)

Sample answer:

A diagonal divides the parallelogram into two triangles. If I can prove that the two triangles are congruent, then the corresponding angles and corresponding sides of those two triangles must be congruent. To prove triangle congruency, I can use the properties of the angles formed when parallel lines are cut by a transversal.

  1. Diagonals of Parallelograms

You will use GeoGebra to explore the diagonals of parallelogram. Open congruent parts of parallelograms again, and complete each step below.

  1. Draw diagonals and on parallelogram ABCD. Make a point where the diagonals intersect, and label it E. Measure and record the lengths of the four line segments you just created.

Sample answer:

Line segment / Length
/ 2.5
/ 1.5
/ 2.5
/ 1.5
  1. Based on your measurements in part a, what is the relationship between the diagonals of parallelogram ABCD?

Sample answer:

Point E divides each diagonal in half. It is the midpoint of and . This means the two diagonals bisect each other at point E.

  1. Move vertices A, B, and C to modify the original parallelogram. As you change the parallelogram, notice what happens to the diagonals. How does moving the vertices affect the relationship between the diagonals that you noted in part b?

Sample answer:

Moving the vertices does not affect the relationship between the diagonals. They bisect each other in all cases.

  1. Explain how you can prove your observations in part b using a traditional proof. You don’t need to develop the steps of the proof; simply describe an approach that you might use. Which theorems might appear in your proof? (Hint: Use the two diagonals of the parallelogram to assist you in describing an approach.)

Sample answer:

The two diagonals divide the parallelogram into four triangles. If I can prove that the two triangles opposite one another are congruent, then the corresponding sides of those two triangles must be congruent. These corresponding sides will help show that the lengths of the line segments on either side of the point of intersection are equal, proving bisection. To prove triangle congruency, I can use the properties of the angles formed when parallel lines are cut by a transversal and that opposite sides of a parallelogram are congruent.

  1. Now go to diagonals of parallelograms. You’ll see two line segments, and marked with their midpoints, E and F. Verify that E and F divide the line segments equally by measuring and recording the length of the four line segments that you see.

Sample answer:

Line segment / Length
/ 4
/ 4
/ 2.24
/ 2.24
  1. Move and so that their midpoints coincide. Then draw a polygon through points A, B, C, and D to form quadrilateral ABCD. Measure and record the slope of each side of quadrilateral ABCD.

Sample answer:

Line segment / Slope (m)
/ 0.67
/ -0.4
/ 0.67
/ -0.4
  1. Based on your slope measurements in part f, what can you conclude about quadrilateral ABCD?

Sample answer:

The opposite sides of quadrilateral ABCD have equal slopes, which means the opposite sides are parallel. Therefore, by definition, quadrilateral ABCD is a parallelogram.

  1. Suppose that you repeated parts e and f using two line segments of your choice. The line segments could be any length and in any orientation as long as the midpoints were marked correctly and coincided with each other. Would you reach the same conclusion that you reached in part g? How does your conclusion relate to the diagonals of a parallelogram?

Sample answer:

Yes, I would reach the same conclusion. The line segments that I’m drawing represent the diagonals of a parallelogram. If the intersection point of the diagonals divides each diagonal in half, then the quadrilateral that belongs to the diagonals is a parallelogram.

  1. Rectangles

You will use GeoGebra to study the properties of diagonals of a rectangle. Open rectangles, and complete each step below.

  1. Draw diagonals and on rectangle ABCD. Measure and record the lengths of the diagonals.

Sample answer:

Diagonal / Length
/ 7.21
/ 7.21
  1. How do the measurements of the diagonals compare?

Sample answer:

and are equal in length.

  1. Move vertices A, B, and C to modify the original rectangle. As you change the rectangle, notice what happens to the diagonals. How does moving the vertices affect the relationship between the diagonals that you noted in part b?

Sample answer:

Moving the vertices does not affect the relationship between the lengths. The two diagonals remain equal in length.

  1. Explain how you can prove your observations in part c using a traditional proof. You don’t need to develop the steps of the proof; simply describe an approach that you might use. Which theorems might appear in your proof? (Hint: Use the two diagonals of the parallelogram to assist you in describing an approach.)

Sample answer:

Rectangles are parallelograms. By definition, opposite sides are congruent and all four angles have the same measure. Drawing the two diagonals of a rectangle divides the rectangle into triangles. If I choose a pair of triangles that each have a different diagonal for a side, I can prove them to be congruent. If the triangles are congruent, the corresponding sides (diagonals of the rectangle in this case) are also congruent.

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