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2.16: Parallelogram Proofs

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    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

    f-d_b44f3f753ae5d2c0f64fe05ea27747fece73e8409e7a42a40ebe4a1b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    then

    f-d_994d6497cbae0e45c57daa4d03e6067e99015df4ccb97c8bf5a14596+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

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

    If

    f-d_2118168440f8fc1a13c1238ef26d6fa54c1173fb4e7e4e2a20c7199c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    then

    f-d_528830801800a01007e868c589b2ebf40ab26919f4557be5e605481a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

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

    If

    f-d_46c004c44a75296d2eac72bd737291bae01ab65e1820b5204f3586ca+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    then

    f-d_c6443e1c10cf2e9c4d75ba5b555b815d0a95690ca23ea976a0b40e5f+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

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

    If

    f-d_e46203f33a4881541307b95407d5031345edcb82ae45185ea0c3aa76+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    then

    f-d_7d8fcf2fea2284750352c44fdde20db6255d6298f2cf47be907ff997+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    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.

    f-d_481247021bd2921618edf2f5e05543ff7668412a6a797825b7b69f29+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)

    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?

    f-d_57be9d320081b376883ec8cad8136dedf5e1644f34d20438c1632327+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)

    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.

    f-d_481247021bd2921618edf2f5e05543ff7668412a6a797825b7b69f29+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)

    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?

    f-d_b1740b63357c5d41fd536f61ac4336ce081028133139b46445b5a7af+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{12}\)

    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?

    f-d_ad6c14bc1b86d3eeeaa76acbda23277a9070ebd2c788dea374e53bbc+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{13}\)

    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.


    1. f-d_f115e63a62ea93db5c7c2626104f16d483f975e8fb7d2ce76dce2651+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    2. f-d_d4552bb5eb475c4c4ce3f5b1bbcbc5a529242c4e35b20e4d16b53475+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    3. f-d_6e35e37accd9e3dfbd07885a830ad68194d5e01d384acf20f8b48d8e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)
    4. f-d_24cfeae21b610c1ef10b4fb97b5212b20d60593b74aa485a85d1a5cd+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    5. f-d_44353c64a9f78624e106f20154cd9f99181debe964e91b4047f2973c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    6. f-d_a872533899b5e49c28e959cd6bdcf8536a59b66b67d863c5d332452d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    7. f-d_f099d07dd39fc5387cb4b57ef6718c22cca19fc3bada97ce191a6d3c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{20}\)
    8. f-d_693a8ec5c82d2b046ae574582eb1e6f1d7f4a999dfab808061a26406+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)
    9. f-d_aa10e238732f18cd00acc1324d6bc0b283aa6be2ee92559b3fb1718d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{22}\)
    10. f-d_1e7c0639f04b1522952e96b432d4ef96930fc55563a7f0762c96643f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{23}\)
    11. f-d_098fceba94a4685311ba9bbd93bde184c39b57bef6c4f493e4b25f03+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{24}\)
    12. f-d_3c7e18be2f36809db07c1f98cd412b8c452bc11585c76408ea1ed6be+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{25}\)

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


    1. f-d_1539ca43a25a0987c6ec6a5c3bf136b42087fe789e8b551ba121479e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{26}\)
    2. f-d_f779028e7d8b015f18326b4aa28e0043d1270e0aaaa7253c6282e096+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{27}\)
    3. f-d_82c1c140c26b3569c7f61c021a892308a80816efd2c32dc79a777ed5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{28}\)
    4. f-d_cbd0d4db01f59e53f702d0678534bf3c369bd54ac751831222fd6e13+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{29}\)
    5. f-d_ef3d72d5ae275cfd7f794254471af20f5978e7818d4571316bd86f79+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{30}\)
    6. f-d_9e4b4d0af7226ede7751c9928b9412b04996fb55808ad75983dd3319+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{31}\)

    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
    f-d_14c5c7af431574174f61e564b4ce1c17a9f6dafef030e565b85454c6+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{32}\)

    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.
    1. Parallelogram Diagonals Theorem Converse
    f-d_4c50cdd807989540195ffb2226df7dc09513ff33267308d9f8203ca8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{33}\)

    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.
    1. Given: \(\angle ADB\cong \angle CBD\), \(\overline{AD}\cong \overline{BC}\)

    Prove: \(ABCD\) is a parallelogram

    f-d_481247021bd2921618edf2f5e05543ff7668412a6a797825b7b69f29+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{34}\)
    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

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