Type of Class: All levels of high school Geometry

Related VA SOL: G.8

Time Frame: 90-minute block period

Objectives:

·        Students will be able to find relationship among angles, sides, and diagonals of parallelograms.

·        Students will be able to find characteristics of quadrilaterals that indicate that they are parallelograms.

·        Students will know how to solve algebraic and coordinate problems using the properties of parallelograms.

·        Students will know how to solve practical problems using the properties of parallelograms.

Materials:

·        Geometer’s Sketchpad

Procedures:

1.      Students will construct a parallelogram on Geometer’s Sketchpad and make conjectures about the relationship between opposite sides, angles, and diagonals of a parallelogram.

Theorems:[1]

·        Opposite sides of a parallelogram are congruent.

·        Opposite angles of a parallelogram are congruent.

·        The diagonals of a parallelogram bisect each other.

 

2.      Students will use a two-column proof to prove the theorems. For example:

 

 

Given: Parallelogram   

           ABCD

Prove:

           

Statements

1. Parallelogram ABCD

2.  

   

3. Ð1 @ Ð4

    Ð2 @ Ð3

4.  

5. DABC @ DCDA

6.

   

 

Reasons

1. Given

2. Definition of

    Parallelogram

 

3. If ||, then alternate

    interior angles are 

    congruent.

4. Reflexive Property

5. ASA (2,3,4)

6. CPCTC

 

 

3.      Students will use the theorems to solve algebraic problems. For example:

Text Box: Find the value of each variable in parallelogram ABCD. Explain your reasoning.

 


 

 

4.      Students will construct the diagonals of a rhombus, a rectangle and a kite on Geometer’s Sketchpad. They will make conjectures about the diagonals.

Diagonals of Special Quadrilaterals[2]

·        The diagonals of a rhombus are perpendicular.

·        The diagonals of a rectangle are congruent.

·        Exactly one diagonal of a kite is a line of symmetry for the kite and the perpendicular bisector of the other diagonal.

 

5.      Students will use these properties to solve coordinate problems. For example:

Text Box: Prove that the diagonals of the rectangle are congruent.  

 

 

6.      Students will work on applications using the theorems and properties. For instance:

Text Box: Your grandmother made you a beautiful quilt for your 16th birthday. The quilt is divided by its diagonals. If AE = 10 units, find BC.  

 

 

7.      Another example: Although traditional houses of the Kpelle people of Liberia are circular, modern Kpelle structures are rectangular. To determine the space that a building will occupy, the Kpelle sometimes place two long wooden sticks across each other as the diagonals of the rectangle. How must the lengths of the sticks be related in order to produce a rectangular structure?[3]

 

[Jorwah Town Photo]


 

[1] Aichele, D.B., Hopfensperger, P.W., Leiva, M.A., Mason, M.M., Murphy, S.J., Schell, V.J., Vheru, M.C. (1998). Geometry: Exploration and application. Evanston, IL:  McDougal Littell, 346-352.

[2] Aichele, D.B., Hopfensperger, P.W., Leiva, M.A., Mason, M.M., Murphy, S.J., Schell, V.J., Vheru, M.C. (1998). Geometry: Exploration and application. Evanston, IL:  McDougal Littell, 347-352.

[3] Aichele, D.B., Hopfensperger, P.W., Leiva, M.A., Mason, M.M., Murphy, S.J., Schell, V.J., Vheru, M.C. (1998). Geometry: Exploration and application. Evanston, IL:  McDougal Littell, 357.