### Quadrilaterals that are Parallelograms

Recall that a parallelogram is a quadrilateral with two pairs of parallel sides. Even if a quadrilateral is not marked with having two pairs of sides, it still might be a parallelogram. The following is a list of theorems that will help you decide if a quadrilateral is a parallelogram or not.

1. **Opposite Sides Theorem Converse:** If both pairs of opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.

If then

2. **Opposite Angles Theorem Converse:** If both pairs of opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.

If then

3. **Parallelogram Diagonals Theorem Converse:** If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.

If then

4. **Parallel Congruent Sides Theorem:** If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.

If then

You can use any of the above theorems to help show that a quadrilateral is a parallelogram. If you are working in the

- The Slope Formula,
y2−y1x2−x1 . (Remember that if slopes are the same then lines are parallel). - The Distance Formula,
(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√ . (This will help you to show that two sides are congruent). - The Midpoint Formula,
(x1+x22,y1+y22) . (If the midpoints of the diagonals are the same then the diagonals bisect each other).

What if you were given four pairs of coordinates that form a quadrilateral? How could you determine if that quadrilateral is a parallelogram?

### Examples

#### Example 1

Prove the Parallel Congruent Sides Theorem.

Given:

Prove:

Statement |
Reason |
---|---|

1. |
1. Given |

2. |
2. Alternate Interior Angles |

3. |
3. Reflexive PoC |

4. |
4. SAS |

5. |
5. CPCTC |

6. |
6. Opposite Sides Converse |

#### Example 2

What value of

#### Example 3

Prove the Opposite Sides Theorem Converse.

Given:

Prove:

Statement |
Reason |
---|---|

1. |
1.Given |

2. |
2. Reflexive PoC |

3. |
3. SSS |

4. |
4. CPCTC |

5. |
5. Alternate Interior Angles Converse |

6. |
6. Definition of a parallelogram |

#### Example 4

Is quadrilateral

By the Opposite Angles Theorem Converse,

#### Example 5

Is the quadrilateral

Let’s use the Parallel Congruent Sides Theorem to see if

Next find the slopes to check if the lines are parallel.

### Review

For questions 1-12, determine if the quadrilaterals are parallelograms.

For questions 13-18, determine the value of

For questions 19-22, determine if

A(8,−1),B(6,5),C(−7,2),D(−5,−4) A(−5,8),B(−2,9),C(3,4),D(0,3) A(−2,6),B(4,−4),C(13,−7),D(4,−10) A(−9,−1),B(−7,5),C(3,8),D(1,2)

Fill in the blanks in the proofs below.

*Opposite Angles Theorem Converse*

Given:

Prove:

Statement |
Reason |
---|---|

1. | 1. |

2. |
2. |

3. | 3. Definition of a quadrilateral |

4. |
4. |

5. | 5. Combine Like Terms |

6. | 6. Division PoE |

7. \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary \begin{align*}\angle A\end{align*} and \begin{align*}\angle D\end{align*} are supplementary | 7. |

8. | 8. Consecutive Interior Angles Converse |

9. \begin{align*}ABCD\end{align*} is a parallelogram | 9. |

*Parallelogram Diagonals Theorem Converse*

Given: \begin{align*}\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}\end{align*}

Prove: \begin{align*}ABCD\end{align*} is a parallelogram

Statement |
Reason |
---|---|

1. | 1. |

2. | 2. Vertical Angles Theorem |

3. \begin{align*}\triangle AED & \cong \triangle CEB\\ \triangle AEB & \cong \triangle CED\end{align*} | 3. |

4. | 4. |

5. \begin{align*}ABCD\end{align*} is a parallelogram | 5. |

- Given: \begin{align*}\angle ADB \cong \angle CBD, \overline{AD} \cong \overline{BC}\end{align*} Prove: \begin{align*}ABCD\end{align*} is a parallelogram

Statement |
Reason |
---|---|

1. | 1. |

2. \begin{align*}\overline{AD} \| \overline{BC}\end{align*} | 2. |

3. \begin{align*}ABCD\end{align*} is a parallelogram | 3. |

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4.