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5.9: Parallelograms

  • Page ID
    4993
  • Find unknown angle measurements of quadrilaterals with two pairs of parallel sides.

    A parallelogram is a quadrilateral with two pairs of parallel sides.

    f-d_ff6ff6ce996bf5f0d715a1f7816d666fd4538d7589e5fa43db6b87bf+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    Notice that each pair of sides is marked parallel (for the last two shapes, remember that when two lines are perpendicular to the same line then they are parallel). Parallelograms have a lot of interesting properties.

    Facts about Parallelograms

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

    If

    f-d_6cc6f1714d92491c4dc1108f4513dd757386dd171c6c30bcc62dbb3a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    then

    f-d_368d168609dc045459d5e70faa3765b22eca0f6c4f916be35d17e8ae+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

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

    If

    f-d_75f9c402da32018e3a196be527e867e32a2b2732fdb6667b79e501a7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    then

    f-d_38a3a7e2860bd48a289437b11c5d8d38969c1bee2977b985fe32989e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    3. Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then all pairs of consecutive angles are supplementary.

    If

    f-d_75f9c402da32018e3a196be527e867e32a2b2732fdb6667b79e501a7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    then

    f-d_38a3a7e2860bd48a289437b11c5d8d38969c1bee2977b985fe32989e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    \(\begin{aligned} m\angle A+m\angle D&=180^{\circ} \\ m\angle A+m\angle B&=180^{\circ} \\ m\angle B+m\angle C&=180^{\circ} \\ m\angle C+m\angle D&=180^{\circ} \end{aligned}\)

    4. Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.

    If

    f-d_75f9c402da32018e3a196be527e867e32a2b2732fdb6667b79e501a7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    then

    f-d_0f402f150856274ce28e896c9006901772509ecee54673eff6843ea6+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)

    What if you were told that \(FGHI\) is a parallelogram and you are given the length of FG\) and the measure of \(\angle F\)? What can you determine about \(HI\), \(\angle H\), \(\angle G\), and \(\angle I\)?

    Example \(\PageIndex{1}\)

    Show that the diagonals of \(FGHJ\) bisect each other.

    f-d_03a8e7d7267d582bcda6e32549da3b61910abd4aa3983417573f8595+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)

    Solution

    Find the midpoint of each diagonal.

    \(\begin{aligned}\text{ Midpoint of } \overline{FH}:& \left(\dfrac{−4+6}{2}, \dfrac{5−4}{2}\right)=(1,0.5) \\ \text{ Midpoint of } of\: \overline{GJ}: & \left(\dfrac{3−1}{2}, \dfrac{3−2}{2}\tight)=(1,0.5)\end{aligned}\)

    Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.

    Example \(\PageIndex{2}\)

    Find the measures of a and b in the parallelogram below:

    f-d_345c1eb74af71473e0e2933ba2798bd8cc094438b8f45c2dcebd9989+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)

    Solution

    Consecutive angles are supplementary so \(127^{\circ}+m\angle b=180^{\circ}\) which means that \(m\angle b=53^{\circ}\). \(a\) and \(b\) are alternate interior angles and since the lines are parallel (since its a parallelogram), that means that \(m\angle a=m\angle b=53^{\circ}\).

    Example \(\PageIndex{3}\)

    \(ABCD\) is a parallelogram. If \(m\angle A=56^{\circ}\), find the measure of the other angles.

    Solution

    First draw a picture. When labeling the vertices, the letters are listed, in order.

    f-d_c5df69650ecbb0d4710c414d22b150ec22f31e9d7ddb1d0854803dda+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{12}\)

    If \(m\angle A=56^{\circ}\), then \(m\angle C=56^{\circ}\) by the Opposite Angles Theorem.

    \(\begin{aligned}
    &m \angle A+m \angle B=180^{\circ} \quad \text { by the Consecutive Angles Theorem. }\\
    &56^{\circ}+m \angle B=180^{\circ}\\
    &m \angle B=124^{\circ} \quad m \angle D=124^{\circ} \quad \text { because it is an opposite angle to } \angle B
    \end{aligned}\)

    Example \(\PageIndex{4}\)

    Find the values of \(x\) and \(y\).

    f-d_bbd796e60f9972dc63a286a56789a9500fb9e460ff1fa19cbbf0f8b5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{13}\)

    Solution

    Remember that opposite sides of a parallelogram are congruent. Set up equations and solve.

    \(\begin{aligned} 6x−7&=2x+9 \\ 4x&=16 \\ x&=4 \\ y+3&=12 \\ y&=9\end{aligned} \)

    Example \(\PageIndex{5}\)

    Prove the Opposite Sides Theorem.

    f-d_481247021bd2921618edf2f5e05543ff7668412a6a797825b7b69f29+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{14}\)

    Solution

    Given: \(ABCD\) is a parallelogram with diagonal \(\overline{BD}\)

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

    Statement Reason
    1. \(ABCD\) is a parallelogram with diagonal \(\overline{BD}\) 1. Given
    2. \(\overline{AB}\parallel \overline{DC}\), \(\overline{AD}\parallel \overline{BC}\) 2. Definition of a parallelogram
    3. \(\angle ABD\cong \angle BDC\), \(\angle ADB\cong \angle DBC\) 3. Alternate Interior Angles Theorem
    4. \(\overline{DB}\cong \overline{DB}\) 4. Reflexive PoC
    5. \(\Delta ABD\cong \Delta CDB\) 5. ASA
    6. \(\overline{AB}\cong \overline{DC}\), \(\overline{AD}\cong \overline{BC}\) 6. CPCTC

    The proof of the Opposite Angles Theorem is almost identical.

    Review

    \(ABCD\) is a parallelogram. Fill in the blanks below.

    f-d_4c50cdd807989540195ffb2226df7dc09513ff33267308d9f8203ca8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{15}\)
    1. If \(AB=6\), then \(CD= \text{______}\).
    2. If \(AE=4\), then \(AC= \text{______}\).
    3. If \(m\angle ADC=80^{\circ}\), \(m\angle DAB = \text{______}\).
    4. If \(m\angle BAC=45^{\circ}\), \(m\angle ACD = \text{______}\).
    5. If \(m\angle CBD=62^{\circ}\), \(m\angle ADB = \text{______}\).
    6. If \(DB=16\), then \(DE = \text{______}\).
    7. If\( m\angle B=72^{\circ}\) in parallelogram \(ABCD\), find the other three angles.
    8. If \(m\angle S=143^{\circ}\) in parallelogram \(PQRS\), find the other three angles.
    9. If \(\overline{AB}\perp \overline{BC}\) in parallelogram \(ABCD\), find the measure of all four angles.
    10. If \(m\angle F=x^{\circ}\) in parallelogram \(EFGH\), find the other three angles.

    For questions 11-18, find the values of the variable(s). All the figures below are parallelograms.

    1. f-d_ac67dd022466e9db02576fdca241d797010ed4fc03b3fe649b2f0b1f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)
    2. f-d_a32e2c3617a37efd77e0d013e3f4788789075409559f44092fb8ff01+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    3. f-d_a973128a397929358d8b4179e3bd7548dcb85fefdd58e86bf3533fec+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    4. f-d_a2e291c76568c99b080cd4707788ff1d9dc6d223aae3be872d751f2c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    5. f-d_a209b5cd1cf9d73521e65993d910557d2c2f9708605c92a30986abd7+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{20}\)
    6. f-d_98662387d16e923a2f946322721d102169628550b15a697988267d74+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)
    7. f-d_9680af28a174e6a54198a609c73189aae8f2fd57d2dd8725805de7fd+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{22}\)
    8. f-d_ad9e5078b4dc578e2daaed9eadf826bb808145e64862f78f5713e7fc+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{23}\)

    Use the parallelogram \(WAVE\) to find:

    f-d_4e8938772296e11bae4816b25c00cd47a8e5e0b3ea528676c779637a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{24}\)
    1. \(m\angle AWE\)
    2. \(m\angle ESV\)
    3. \(m\angle WEA\)
    4. \(m\angle AVW\)

    Find the point of intersection of the diagonals to see if \(EFGH\) is a parallelogram.

    1. \(E(−1,3), F(3,4), G(5,−1), H(1,−2)\)
    2. \(E(3,−2), F(7,0), G(9,−4), H(5,−4)\)
    3. \(E(−6,3), F(2,5), G(6,−3), H(−4,−5)\)
    4. \(E(−2,−2), F(−4,−6), G(−6,−4), H(−4,0)\)

    Fill in the blanks in the proofs below.

    1. Opposite Angles Theorem
    f-d_481247021bd2921618edf2f5e05543ff7668412a6a797825b7b69f29+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{25}\)

    Given: \(ABCD\) is a parallelogram with diagonal \(\overline{BD}\)

    Prove: \(\angle A\cong \angle C\)

    Statement Reason
    1. 1. Given
    2. \(\overline{AB}\parallel \overline{DC}\), \(\overline{AD}\parallel \overline{BC}\) 2.
    3. 3. Alternate Interior Angles Theorem
    4. 4. Reflexive PoC
    5. \(\Delta ABD\cong \Delta CDB\) 5.
    6. \(\angle A\cong \angle C\) 6.
    1. Parallelogram Diagonals Theorem
    f-d_4c50cdd807989540195ffb2226df7dc09513ff33267308d9f8203ca8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{26}\)

    Given: \(ABCD\) is a parallelogram with diagonals \(\overline{BD}\) and \(\overline{AC}\)

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

    Statement Reason
    1. 1.
    2. 2. Definition of a parallelogram
    3. 3. Alternate Interior Angles Theorem
    4. \(\overline{AB}\cong \overline{DC}\) 4.
    5. 5.
    6.\( \overline{AE}\cong \overline{EC}\),\(\overline{DE}\cong \overline{EB}\) 6.
    1. Find \(x\), \(y^{\circ}\), and \(z^{\circ}\). (The two quadrilaterals with the same side are parallelograms.)
    f-d_af556799a2227aecf648002f19e53209b290e583f53366f957f841e9+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{27}\)

    Vocabulary

    Term Definition
    parallelogram 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.

    Additional Resources

    Interactive Element

    Video: Parallelograms Principles - Basic

    Activities: Parallelograms Discussion Questions

    Study Aids: Parallelograms Study Guide

    Practice: Parallelograms

    Real World: Parallelograms