Concept Byte: Paper-Folding Conjectures

Use With Lesson 4-5

ACTIVITY

Isosceles triangles have two congruent sides. Folding one of the sides onto the other will suggest another important property of isosceles triangles.

Activity 1

• Step 1 Construct an isosceles Δ A B C cap delta eh b c  on tracing paper, with A C ¯ ≅ B C ¯ . eh c bar , approximately equal to . b c bar , .

  

• Step 2 Fold the paper so the two congruent sides fit exactly one on top of the other. Crease the paper. Label the intersection of the fold line and A B ¯ eh b bar  as point D.

1. What do you notice about ∠ A angle eh  and ∠ B ? angle b question mark  Compare your results with others. Make a conjecture about the angles opposite the congruent sides in an isosceles triangle.
2. 1. Study the fold line C D ¯ c d bar  and the base A B ¯ . eh b bar , .  What type of angles are ∠ C D A angle c d eh  and ∠ C D B ? angle c d b question mark  How do A D ¯ eh d bar  and B D ¯ b d bar  seem to be related?
   2. Use your answers to part (a) to complete the conjecture: The fold line C D ¯ c d bar  is the __?__ of the base A B ¯ eh b bar  of isosceles Δ A B C . cap delta eh b c .
   

Activity 2

In Activity 1, you made a conjecture about angles opposite the congruent sides of a triangle. You can also fold paper to study whether the converse is true.

• Step 1 On tracing paper, draw acute angle F and one side F G ¯ . f g bar , .  Construct ∠ G angle g  as shown, so that ∠ G ≅ ∠ F . angle g approximately equal to angle f .
• Step 2 Fold the paper so ∠ F angle f  and ∠ G angle g  fit exactly one on top of the other.

3. Why do sides 1 and 2 meet at point H on the fold line? Make a conjecture about sides F H ¯ f h bar  and G H ¯ g h bar  opposite congruent angles in a triangle.
4. Write your conjectures from Questions 1 and 3 as a biconditional.

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