7.3 Identifying and Verifying Parallelograms

DEFINITION

A plane convex quadrilateral is a parallelogram, if and only if, it has[b] opposite parallel sides[/b]. In the following figure AB//CD (AB is parallel to CD) and AC//BD (AC is parallel to BD).

Parallelogram

Property 1

Opposite congruent angles.[br]

Analysis

Change points A, B or C. What do you notice?

ARGUMENT

Exercise 1

[justify]Research and prove the argument of following property: "Every convex quadrilateral that has congruent opposite angles is a parallelogram" (Remember that the sum of the internal angles of the quadrilateral is 360º).[/justify][br]

Property 2

Congruent opposite sides.

Analysis

Change points A, B or C. What do you notice?

Exercise 2:

[justify]Research and prove the argument of following property: "Every convex quadrilateral that has opposing congruent sides is a parallelogram" (Note: Draw a diagonal and use triangle congruence). [/justify]

Diagonals of the Paralellogram

Analysis

Change vertices A, B or C. What do you notice?

Proof of the Property

EXERCISE 3

[justify]Research and prove the argument of following property: "Every convex quadrilateral is a parallelogram, when their diagonals intersect at midpoints."[/justify]

Move points A, B or C.

Exercise 4

Move points A, B or C of the previous figure. What can you notice?[br][br]

A, B, C e D are always the vertices of a rhombus. A, B, C e D are always the vertices of a square. A, B, C e D are the vertices of a parallelogram. If two line segments intersect at the respective midpoints, then the points of intersection are vertices of the parallelogram.

7.3 Identifying and Verifying Parallelograms

DEFINITION

Parallelogram

Property 1

Analysis

ARGUMENT

Exercise 1

Property 2

Analysis

Exercise 2:

Diagonals of the Paralellogram

Analysis

Proof of the Property

EXERCISE 3

Move points A, B or C.

Exercise 4

Information: 7.3 Identifying and Verifying Parallelograms