2.9: Proving Quadrilaterals are Parallelograms
Learning Objectives
- Define converse and write the converse of given statements.
- Evaluate a converse statement.
- Prove a quadrilateral is a parallelogram given: two pairs of parallel opposite sides; or, two pairs of congruent opposite sides; or, two pairs of congruent opposite angles; or, two pairs of supplementary consecutive angles; or, diagonals that bisect each other.
- Also, prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.
Converse Statements
A converse statement reverses the order of the hypothesis and conclusion in an if-then conditional statement, and is only sometimes true.
Conditional (if-then) Statement:
Converse Statement:
A converse reverses the ________________________ and the _______________________.
Consider the statement: “If you live in Los Angeles, then you live in California.”
To write the converse of this statement, switch the hypothesis and conclusion:
The converse statement would be “If you live in California, then you live in Los Angeles.” Can you see that this statement is not true? There are lots of people who live in California who do not live in Los Angeles. Some counterexamples to this statement are: people who live in Sacramento, people who live in San Jose, people who live in Oakland, etc.
A counterexample is an example that proves that a statement is ___________________.
An example of a statement that is true and whose converse is also true is as follows:
If it is 10:00, then it was 9:00 an hour ago.
The converse of this statement is “If it was 9:00 an hour ago, then it is 10:00.”
This converse is true.
All geometric definitions have true converses.
For example, the definition of a triangle could be written this way:
If a shape is a triangle, then it is a three-sided polygon.
Because this statement is a definition, its converse is also true.
If a shape is a three-sided polygon, then it is a triangle.
- Geometric _____________________________ have converses that are true.
Reading Check:
1. Write the converse of the following statement:
If an animal is a cat, then it has a tail.
2. Is the converse you wrote in question #1 true? If not, provide a counterexample that proves it is false.
Proving a Quadrilateral is a Parallelogram
There are 6 ways to prove that a quadrilateral is a parallelogram.
Five of these ways are converses of statements you have already learned.
Fill in the table below with the converses of the definition and theorems you have already learned about parallelograms.
Statements About Parallelograms | Converse (used to prove a quadrilateral is a parallelogram) | |
---|---|---|
Definition | If a quadrilateral is a parallelogram, then both pairs of opposite sides are parallel. | |
Theorems about parallelograms | If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. | |
Theorems about parallelograms | If a quadrilateral is a parallelogram, then two pairs of opposite angles are congruent. | |
Theorems about parallelograms | If a quadrilateral is a parallelogram, then two pairs of consecutive angles are supplementary. | |
Theorems about parallelograms | If a quadrilateral is a parallelogram, then diagonals bisect each other. | |
New theorem | If a quadrilateral is a parallelogram, then one pair of sides is both parallel and congruent. |
Reading Check:
Which statement could be used to prove that each of the following shapes is a parallelogram?
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