Topics
Congruence and Similarity
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India
Class IX

Congruent Triangles (Investigation)

Lesson
Lesson

Objectives

  • To explore what makes triangles congruent.
  • To explore why some methods cannot be used to prove congruence.

Materials

  • 3 Different colored pipe cleaners (at least 2 of each color)
  • Scissors
  • Tape
  • Protractor
  • Paper
  • Pencil

Procedure

  1. Use your scissors to cut the first pipe cleaner color into pieces that are 4 inches long.
  2. Use your scissors to cut the second pipe cleaner color into pieces that are 5 inches long.
  3. Use your scissors to cut the last pipe cleaner color into pieces that are 6 inches long.
  4. Follow the instructions on creating the triangles and answer the questions that go along with them. Every time you create a new triangle draw a picture of it and record the lengths of each side as well as the angle if applicable.

Questions

  1. Create a triangle using one pipe cleaner of each color. How can you create a triangle that is congruent to the one you have just created? What rule for congruency does this embody?
  2. Fix an angle between two pipe cleaner pieces of your choice. Once you have the angle tape down the pieces so that they do not move. Use your protractor to measure the angle. Complete the triangle with a third piece of pipe cleaner. How can you create a triangle that is congruent to the one that you have just created? What rule for congruency does this embody?
  3. Can you think of another triangle that would be congruent to the first triangle you created in question 2? What rule of congruency did you use to create it?
  4. Compare the congruent triangles you came up with in numbers 1, 2, and 3 with the congruent triangles someone else came up with for those numbers. How are they the same? How are they different? Are they congruent to each other? Why or why not?
  5. Use any three pieces of pipe cleaner to create a new triangle. Can you create a triangle that is congruent to your triangle by using SSA (side-side-angle)? If yes, are all of the triangles that you can create using SSA congruent to your triangle? Why or why not?
  6. Can you use AAA to create a triangle that is congruent to your triangle? If yes, are all of the triangles that you can create using SSA congruent to your triangle? Why or why not?
  7. From what you have investigated, what are the ways that you can use to prove congruence of two triangles?

Outcomes

9.G.T.1

Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

9.G.T.2

Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. The angles opposite to equal sides of a triangle are equal. The angles opposite to equal sides of a triangle are equal. Triangle inequalities and relation between ‘angle and facing side’; inequalities in a triangle.

9.G.Q.1

The diagonal divides a parallelogram into two congruent triangles. In a parallelogram opposite sides are equal and conversely. In a parallelogram opposite angles are equal and conversely. A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. In a parallelogram, the diagonals bisect each other and conversely.

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