Jump Start
What is wrong with this proof?
Given: Quadrilateral
Prove: The diagonal
Draw diagonal
Therefore,
Learning Focus
Use theorems about the relationships of angles formed by parallel lines and a transversal to prove properties of parallelograms.
Previously, we made several conjectures about properties of special types of parallelograms: rhombuses, rectangles, and squares. How might we prove that our conjectures about parallelograms are true?
How can we prove a quadrilateral is a parallelogram if we don’t know the opposite sides are parallel? That is, what other characteristics might define a parallelogram?
Open Up the Math: Launch, Explore, Discuss
1.
Explain how you would locate the center of rotation for the following parallelogram. What convinces you that the point you have located is the center of rotation?
Previously, you have made conjectures about properties of parallelograms based on identifying line symmetry and rotational symmetry for various types of parallelograms. Now that we have additional knowledge about the angles formed when parallel lines are cut by a transversal, and we have criteria for convincing ourselves that two triangles are congruent, we can more formally prove some of the things we have noticed about parallelograms experimentally.
2.
If you haven’t already, draw one or both of the diagonals in the parallelogram. Use this diagram to prove this statement: Opposite sides of a parallelogram are congruent.
3.
If you haven’t already, draw one or both of the diagonals in the parallelogram. Use this diagram to prove this statement: Opposite angles of a parallelogram are congruent.
4.
Use this diagram to prove this statement: The diagonals of a parallelogram bisect each other.
Ready for More?
The statements we have proved above extend our knowledge of properties of all parallelograms beyond the definition of a parallelogram. That is, not only are the opposite sides parallel, they are also congruent; opposite angles are congruent; and the diagonals of a parallelogram bisect each other. A parallelogram has
If we have a quadrilateral that has some of these properties, can we convince ourselves that the quadrilateral is a parallelogram? How many of these properties do we need to know before we can conclude that a quadrilateral is a parallelogram?
In the Ready for More of Justification and Proof, you proved the following theorems:
If alternate interior angles formed by two lines and a transversal are congruent, then the lines are parallel.
If corresponding angles formed by two lines and a transversal are congruent, then the lines are parallel.
These theorems are useful in proving that quadrilaterals with given features are parallelograms.
Consider the following statements. If you think the statement is true, create a diagram and write a convincing argument to prove the statement.
a.
If opposite sides and opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
b.
If opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
c.
If opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
d.
If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
Takeaways
Previously, we surfaced many conjectures about parallelograms by experimenting and reasoning with rigid transformations. Today, we formally proved the following theorems about parallelograms:
We also considered some ways to show that a quadrilateral is a parallelogram, by using converse statements, such as:
We used both the definitions of rigid transformations and triangle congruence criteria in our proofs, as well as postulates or theorems about parallel lines.
Rigid transformations are most useful when
Triangle congruence criteria are most useful when
The parallel postulates and theorems about relationships among angles formed by parallel lines and a transversal are most useful when
Lesson Summary
In this lesson, we drew upon our understanding of rigid transformations, triangle congruence criteria, and postulates and theorems about parallel lines to prove many of our conjectures about the sides, angles, and diagonals of parallelograms.
Retrieval
1.
Use a compass and straightedge to construct an equilateral triangle inscribed in the circle.
Determine if the following statements are true or false.
2.
All rectangles are squares.
3.
Equilateral triangles are also isosceles.
4.
When a pentagon goes through a translation, the sides of the pre-image and image will be parallel to one another.
5.
Diagonals in quadrilaterals always bisect the angles.
