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Parallelograms

Quadrilateral with two pairs of parallel sides.

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Parallelograms

Parallelograms

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

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 then

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

If then

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

If then

\begin{align*}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{align*}

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

If then

What if you were told that \begin{align*}FGHI\end{align*} is a parallelogram and you are given the length of \begin{align*}FG\end{align*} and the measure of \begin{align*}\angle F \end{align*}? What can you determine about \begin{align*}HI\end{align*}, \begin{align*}\angle H \end{align*}, \begin{align*}\angle G \end{align*}, and \begin{align*}\angle I \end{align*}?

  

 

Examples

Example 1

Show that the diagonals of \begin{align*}FGHJ\end{align*} bisect each other.

Find the midpoint of each diagonal.

\begin{align*}&\text{Midpoint of} \ \overline{FH}: \qquad \left ( \frac{-4 + 6 }{2}, \frac{5 - 4}{2} \right ) = (1, 0.5)\\ &\text{Midpoint of} \ \overline{GJ}: \qquad \left ( \frac{3 - 1}{2}, \frac{3 - 2}{2} \right ) = (1,0.5)\end{align*}

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

Example 2

Find the measures of \begin{align*}a\end{align*} and \begin{align*}b\end{align*} in the parallelogram below:

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

Example 3

\begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}m \angle A = 56^\circ\end{align*}, find the measure of the other angles.

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

If \begin{align*}m \angle A = 56^\circ\end{align*}, then \begin{align*}m \angle C = 56^\circ\end{align*} by the Opposite Angles Theorem.

\begin{align*}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{align*}

Example 4

Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

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

\begin{align*}6x - 7 & = 2x + 9 && y + 3 = 12\\ 4x & = 16 && \qquad y = 9\\ x & = 4\end{align*}

Example 5

Prove the Opposite Sides Theorem.

Given: \begin{align*}ABCD\end{align*} is a parallelogram with diagonal \begin{align*}\overline{BD}\end{align*}

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

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

The proof of the Opposite Angles Theorem is almost identical.

Review

\begin{align*}ABCD\end{align*} is a parallelogram. Fill in the blanks below.

  1. If \begin{align*}AB = 6\end{align*}, then \begin{align*}CD =\end{align*} ______.
  2. If \begin{align*}AE = 4\end{align*}, then \begin{align*}AC =\end{align*} ______.
  3. If \begin{align*}m \angle ADC = 80^\circ, m \angle DAB\end{align*} = ______.
  4. If \begin{align*}m \angle BAC = 45^\circ, m \angle ACD\end{align*} = ______.
  5. If \begin{align*}m \angle CBD = 62^\circ, m \angle ADB\end{align*} = ______.
  6. If \begin{align*}DB = 16\end{align*}, then \begin{align*}DE\end{align*} = ______.
  7. If \begin{align*}m \angle B = 72^\circ\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the other three angles.
  8. If \begin{align*}m \angle S = 143^\circ\end{align*} in parallelogram \begin{align*}PQRS\end{align*}, find the other three angles.
  9. If \begin{align*}\overline{AB} \perp \overline{BC}\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the measure of all four angles.
  10. If \begin{align*}m \angle F = x^\circ\end{align*} in parallelogram \begin{align*}EFGH\end{align*}, find the other three angles.

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

Use the parallelogram \begin{align*}WAVE\end{align*} to find:

  1. \begin{align*}m \angle AWE\end{align*}
  2. \begin{align*}m \angle ESV\end{align*}
  3. \begin{align*}m \angle WEA\end{align*}
  4. \begin{align*}m \angle AVW\end{align*}

Find the point of intersection of the diagonals to see if \begin{align*}EFGH\end{align*} is a parallelogram.

  1. \begin{align*}E(-1, 3), F(3, 4), G(5, -1), H(1, -2)\end{align*}
  2. \begin{align*}E(3, -2), F(7, 0), G(9, -4), H(5, -4)\end{align*}
  3. \begin{align*}E(-6, 3), F(2, 5), G(6, -3), H(-4, -5)\end{align*}
  4. \begin{align*}E(-2, -2), F(-4, -6), G(-6, -4), H(-4, 0)\end{align*}

Fill in the blanks in the proofs below.

  1. Opposite Angles Theorem

Given: \begin{align*}ABCD\end{align*} is a parallelogram with diagonal \begin{align*}\overline{BD}\end{align*}

Prove: \begin{align*}\angle A \cong \angle C\end{align*}

Statement Reason
1. 1. Given
2. \begin{align*}\overline{AB} \| \overline{DC},\overline{AD} \| \overline{BC}\end{align*} 2.
3. 3. Alternate Interior Angles Theorem
4. 4. Reflexive PoC
5. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*} 5.
6. \begin{align*}\angle A \cong \angle C\end{align*} 6.
  1. Parallelogram Diagonals Theorem

Given: \begin{align*}ABCD\end{align*} is a parallelogram with diagonals \begin{align*}\overline{BD}\end{align*} and \begin{align*}\overline{AC}\end{align*}

Prove: \begin{align*}\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}\end{align*}

Statement Reason
1. 1.
2. 2. Definition of a parallelogram
3. 3. Alternate Interior Angles Theorem
4. \begin{align*}\overline{AB} \cong \overline{DC}\end{align*} 4.
5. 5.
6. \begin{align*}\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}\end{align*} 6.
  1. Find \begin{align*}x, y^\circ,\end{align*} and \begin{align*}z^\circ\end{align*}. (The two quadrilaterals with the same side are parallelograms.)

 

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.3.

Vocabulary

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.

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