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## Ways to show if a quadrilateral has two pairs of parallel sides.

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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 \begin{align*}x-y\end{align*} plane, you might need to know the formulas shown below to help you use the theorems.

• The Slope Formula, \begin{align*}\frac{y_2 - y_1}{x_2 - x_1}\end{align*}. (Remember that if slopes are the same then lines are parallel).
• The Distance Formula, \begin{align*}\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\end{align*}. (This will help you to show that two sides are congruent).
• The Midpoint Formula, \begin{align*}\left ( \frac{x_1 + x_2 }{2} , \frac{y_1 + y_2}{2} \right )\end{align*}. (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: \begin{align*}\overline{AB} \| \overline{DC}\end{align*}, and \begin{align*}\overline{AB} \cong \overline{DC}\end{align*}

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

Statement Reason
1. \begin{align*}\overline{AB} \| \overline{DC}\end{align*}, and \begin{align*}\overline{AB} \cong \overline{DC}\end{align*} 1. Given
2. \begin{align*}\angle ABD \cong \angle BDC\end{align*} 2. Alternate Interior Angles
3. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} 3. Reflexive PoC
4. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*} 4. SAS
5. \begin{align*}\overline{AD} \cong \overline{BC}\end{align*} 5. CPCTC
6. \begin{align*}ABCD\end{align*} is a parallelogram 6. Opposite Sides Converse

#### Example 2

What value of \begin{align*}x\end{align*} would make \begin{align*}ABCD\end{align*} a parallelogram?

\begin{align*}\overline{AB} \| \overline{DC}\end{align*}. By the Parallel Congruent Sides Theorem, \begin{align*}ABCD\end{align*} would be a parallelogram if \begin{align*}AB = DC\end{align*}.

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

#### Example 3

Prove the Opposite Sides Theorem Converse.

Given: \begin{align*}\overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}\end{align*}

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

Statement Reason
1. \begin{align*}\overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}\end{align*} 1.Given
2. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} 2. Reflexive PoC
3. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*} 3. SSS
4. \begin{align*}\angle ABD \cong \angle BDC, \angle ADB \cong \angle DBC\end{align*} 4. CPCTC
5. \begin{align*}\overline{AB} \| \overline{DC}, \overline{AD} \| \overline{BC}\end{align*} 5. Alternate Interior Angles Converse
6. \begin{align*}ABCD\end{align*} is a parallelogram 6. Definition of a parallelogram

#### Example 4

Is quadrilateral \begin{align*}EFGH\end{align*} a parallelogram? How do you know?

By the Opposite Angles Theorem Converse, \begin{align*}EFGH\end{align*} is a parallelogram.

\begin{align*}EFGH\end{align*} is not a parallelogram because the diagonals do not bisect each other.

#### Example 5

Is the quadrilateral \begin{align*}ABCD\end{align*} a parallelogram?

Let’s use the Parallel Congruent Sides Theorem to see if \begin{align*}ABCD\end{align*} is a parallelogram. First, find the length of \begin{align*}AB\end{align*} and \begin{align*}CD\end{align*} 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{16 + 4}\\ & = \sqrt{20} &&= \sqrt{20}\end{align*}

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

\begin{align*}\text{Slope}\ AB = \frac{5 - 3}{-1-3} = \frac{2}{-4} = -\frac{1}{2} \qquad \text{Slope}\ CD = \frac{-2 +4}{2-6} = \frac{2}{-4} = -\frac{1}{2}\end{align*}

\begin{align*}AB = CD\end{align*} and the slopes are the same (implying that the lines are parallel), so \begin{align*}ABCD\end{align*} is a parallelogram.

### Review

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

For questions 13-18, determine the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} that would make the quadrilateral a parallelogram.

For questions 19-22, determine if \begin{align*}ABCD\end{align*} is a parallelogram.

1. \begin{align*}A(8, -1), B(6, 5), C(-7, 2), D(-5, -4)\end{align*}
2. \begin{align*}A(-5, 8), B(-2, 9), C(3, 4), D(0, 3)\end{align*}
3. \begin{align*}A(-2, 6), B(4, -4), C(13, -7), D(4, -10)\end{align*}
4. \begin{align*}A(-9, -1), B(-7, 5), C(3, 8), D(1, 2)\end{align*}

Fill in the blanks in the proofs below.

1. Opposite Angles Theorem Converse

Given: \begin{align*}\angle A \cong \angle C, \angle D \cong \angle B\end{align*}

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

Statement Reason
1. 1.
2. \begin{align*}m \angle A = m \angle C, m \angle D = m \angle B\end{align*} 2.
3. 3. Definition of a quadrilateral
4. \begin{align*}m \angle A + m \angle A + m \angle B + m \angle B = 360^\circ\end{align*} 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.
1. 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.
1. 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.

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