# 2.16: Parallelogram Proofs

- Page ID
- 7188

Apply theorems to show if a quadrilateral has two pairs of parallel sides.

## 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 x−y plane, you might need to know the formulas shown below to help you use the theorems.

- The Slope Formula, \(\dfrac{y_2−y_1}{x_2−x_1}\). (Remember that if slopes are the same then lines are parallel).
- The Distance Formula, \(\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}\). (This will help you to show that two sides are congruent).
- The Midpoint Formula, \( ( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} )\). (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?

Example \(\PageIndex{1}\)

Prove the Parallel Congruent Sides Theorem.

__Given__: \(\overline{AB}\parallel\overline{DC}\), and \(\overline{AB}\cong \overline{DC}\)

__Prove__: \(ABCD\) is a parallelogram

**Solution**

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

1. \(\overline{AB}\parallel\overline{DC}\), and \(\overline{AB}\cong \overline{DC}\) | 1. Given |

2. \(\angle ABD\cong \angle BDC\) | 2. Alternate Interior Angles |

3. \(\overline{DB}\cong \overline{DB}\) | 3. Reflexive \(PoC\) |

4. \(\Delta ABD\cong \Delta CDB\) | 4. SAS |

5. \(\overline{AD}\cong \overline{BC}\)\) | 5. CPCTC |

6. \(ABCD\) is a parallelogram | 6. Opposite Sides Converse |

Example \(\PageIndex{2}\)

What value of \(x\) would make \(ABCD\) a parallelogram?

**Solution**

\(\overline{AB}\parallel\overline{DC}\). By the Parallel Congruent Sides Theorem, \(ABCD\) would be a parallelogram if \(AB=DC\).

\(\begin{align*} 5x−8 &=2x+13 \\ 3x &=21 \\ x &=7 \end{align*}\)

Example \(\PageIndex{3}\)

Prove the Opposite Sides Theorem Converse.

__Given__: \(\overline{AB}\cong \overline{DC}\), \(\overline{AD}\cong \overline{BC}\)

__Prove__: \(ABCD\) is a parallelogram

**Solution**

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

1. \(\overline{AB}\cong \overline{DC}\),\(\overline{AD}\cong \overline{BC}\) | 1.Given |

2. \(\overline{DB}\cong \overline{DB}\) | 2. Reflexive \(PoC\) |

3. \(\Delta ABD\cong \Delta CDB\) | 3. SSS |

4. \(\angle ABD\cong \angle BDC\),\(\angle ADB\cong \angle DBC\) | 4. \(CPCTC\) |

5. \(\overline{AB}\parallel\overline{DC}\),\(\overline{AD}\parallel\overline{BC}\) | 5. Alternate Interior Angles Converse |

6. \(ABCD\) is a parallelogram | 6. Definition of a parallelogram |

Example \(\PageIndex{4}\)

Is quadrilateral \(EFGH\) a parallelogram? How do you know?

**Solution**

By the Opposite Angles Theorem Converse, \(EFGH\) is a parallelogram.

\(EFGH\) is not a parallelogram because the diagonals do not bisect each other.

Example \(\PageIndex{5}\)

Is the quadrilateral \(ABCD\) a parallelogram?

**Solution**

Let’s use the Parallel Congruent Sides Theorem to see if \(ABCD\) is a parallelogram. First, find the length of AB and CD using the distance formula.

\(\begin{align*} AB &=\sqrt{(−1−3)^2+(5−3)^2} & CD &=\sqrt{(2−6)^2+(−2+4)^2} \\ &=\sqrt{(−4)^{2}+2^2} & &=\sqrt{(−4)^2+2^2} \\ &=\sqrt{16+4}=\sqrt{20} & & =\sqrt{16+4}=\sqrt{20} \end{align*}\)

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

\(\begin{align*} Slope \: AB =\dfrac{5−3}{−1−3} =\dfrac{2}{−4} &=−\dfrac{1}{2} &Slope \: CD=\dfrac{−2+4}{2−6}=\dfrac{2}{−4}=−\dfrac{1}{2} \end{align*}\)

\(AB=CD\) and the slopes are the same (implying that the lines are parallel), so \(ABCD\) is a parallelogram.

## Review

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

For questions 13-18, determine the value of \(x\) and \(y\) that would make the quadrilateral a parallelogram.

For questions 19-22, determine if \(ABCD\) is a parallelogram.

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

Fill in the blanks in the proofs below.

*Opposite Angles Theorem Converse*

__Given__: \(\angle A\cong \angle C\), \(\angle D\cong \angle B\)

__Prove__: \(ABCD\) is a parallelogram

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

1. | 1. |

2. \(m\angle A=m\angle C\), \(m\angle D=m\angle B\) | 2. |

3. | 3. Definition of a quadrilateral |

4. \(m\angle A+m\angle A+m\angle B+m\angle B=360^{\circ}\) | 4. |

5. | 5. Combine Like Terms |

6. | 6. Division \(PoE\) |

7. \(\angle A\) and \(\angle B\) are supplementary \(\angle A\) and \(\angle D\) are supplementary | 7. |

8. | 8. Consecutive Interior Angles Converse |

9. \(ABCD\) is a parallelogram | 9. |

*Parallelogram Diagonals Theorem Converse*

__Given__: \(\overline{AE}\cong \overline{EC}\), \(\overline{DE}\cong \overline{EB}\)

__Prove__: \(ABCD\) is a parallelogram

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

1. | 1. |

2. | 2. Vertical Angles Theorem |

3. \(\Delta AED \cong \Delta CEB\) \(\Delta AEB\cong \Delta CED\) |
3. |

4. | 4. |

5. \(ABCD\) is a parallelogram | 5. |

__Given__: \(\angle ADB\cong \angle CBD\), \(\overline{AD}\cong \overline{BC}\)

__Prove__: \(ABCD\) is a parallelogram

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

1. | 1. |

2. \(\overline{AD}\parallel\overline{BC}\) | 2. |

3. \(ABCD\) is a parallelogram | 3. |

## Review (Answers)

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

## Additional Resources

Interactive Element

Video: Proving a Quadrilateral is a Parallelogram Principles - Basic

Activities: Quadrilaterals that are Parallelograms Discussion Questions

Study Aids: Parallelograms Study Guide

Practice: Parallelogram Proofs

Real World: Quadrilaterals That Are Parallelograms