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Quadrilaterals that are Parallelograms

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

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Quadrilaterals that are Parallelograms

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 xy plane, you might need to know the formulas shown below to help you use the theorems.

  • The Slope Formula, y2y1x2x1. (Remember that if slopes are the same then lines are parallel).
  • The Distance Formula, (x2x1)2+(y2y1)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?




Example 1

Prove the Parallel Congruent Sides Theorem.

Given: AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯, and AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯

Prove: ABCD is a parallelogram

Statement Reason
1. AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯, and AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯ 1. Given
2. ABDBDC 2. Alternate Interior Angles
3. DB¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯ 3. Reflexive PoC
5. AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯ 5. CPCTC
6. ABCD is a parallelogram 6. Opposite Sides Converse

Example 2

What value of x would make ABCD a parallelogram?

AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯. By the Parallel Congruent Sides Theorem, ABCD would be a parallelogram if AB=DC.


Example 3

Prove the Opposite Sides Theorem Converse.

Given: AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯,AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯

Prove: ABCD is a parallelogram

Statement Reason
1. AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯,AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯ 1.Given
2. DB¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯ 2. Reflexive PoC
5. AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯,AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯ 5. Alternate Interior Angles Converse
6. ABCD is a parallelogram 6. Definition of a parallelogram

Example 4

Is quadrilateral EFGH a parallelogram? How do you know?

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

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

Example 5

Is the quadrilateral ABCD a parallelogram?

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.


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

Slope AB=5313=24=12Slope CD=2+426=24=12

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


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.

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

Fill in the blanks in the proofs below.

  1. Opposite Angles Theorem Converse

Given: AC,DB

Prove: ABCD is a parallelogram

Statement Reason
1. 1.
2. mA=mC,mD=mB 2.
3. 3. Definition of a quadrilateral
4. mA+mA+mB+mB=360 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.

Review (Answers)

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

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A quadrilateral with two pairs of parallel sides. A parallelogram may be a rectangle, a rhombus, or a square, but need not be any of the three.

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