What if a college wanted to build two walkways through a parallelogram-shaped courtyard between two buildings? The walkways would be 50 feet and 68 feet long and would be built on the diagonals of the parallelogram with a fountain where they intersect. Where would the fountain be? After completing this Concept, you'll be able to answer questions like this by applying your knowledge of parallelograms.
Watch This
CK-12 Foundation: Chapter6ParallelogramsA
Learn more about parallelogram properties by watching the video at this link.
Guidance
A parallelogram is a quadrilateral with two pairs of parallel sides. Here are some examples:
Notice that each pair of sides is marked parallel. As is the case with the rectangle and square, recall that two lines are parallel when they are perpendicular to the same line. Once we know that a quadrilateral is a parallelogram, we can discover some additional properties.
Investigation: Properties of Parallelograms
Tools Needed: Paper, pencil, ruler, protractor
- Draw a set of parallel lines by placing your ruler on the paper and drawing a line on either side of it. Make your lines 3 inches long.
- Rotate the ruler and repeat this so that you have a parallelogram. Your second set of parallel lines can be any length. If you have colored pencils, outline the parallelogram in another color.
- Measure the four interior angles of the parallelogram as well as the length of each side. Can you conclude anything about parallelograms, other than opposite sides are parallel?
- Draw the diagonals. Measure each and then measure the lengths from the point of intersection to each vertex.
To continue to explore the properties of a parallelogram, see the website:
http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php
In the above investigation, we drew a parallelogram. From this investigation we can conclude:
Opposite Sides Theorem: If a quadrilateral is a parallelogram, then the opposite sides are congruent.
Opposite Angles Theorem: If a quadrilateral is a parallelogram, then the opposite angles are congruent.
Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.
To prove the first three theorems, one of the diagonals must be added to the figure and then the two triangles can be proved congruent.
Proof of 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*} | Given |
2. \begin{align*}\overline{AB} \ || \ \overline{DC}, \ \overline{AD} \ || \ \overline{BC}\end{align*} | Definition of a parallelogram |
3. \begin{align*}\angle ABD \cong BDC, \ \angle ADB \cong DBC\end{align*} | Alternate Interior Angles Theorem |
4. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} | Reflexive PoC |
5. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*} | ASA |
6. \begin{align*}\overline{AB} \cong \overline{DC}, \ \overline{AD} \cong \overline{BC}\end{align*} | CPCTC |
Example A
\begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}m \angle A = 56^\circ\end{align*}, find the measure of the other three angles.
Draw a picture. When labeling the vertices, the letters are listed, in order, clockwise.
If \begin{align*}m \angle A = 56^\circ\end{align*}, then \begin{align*}m \angle C = 56^\circ\end{align*} because they are opposite angles. \begin{align*}\angle B\end{align*} and \begin{align*}\angle D\end{align*} are consecutive angles with \begin{align*}\angle A\end{align*}, so they are both supplementary to \begin{align*}\angle A\end{align*}. \begin{align*}m \angle A+m \angle B=180^\circ, 56^\circ+m \angle B=180^\circ, m \angle B=124^\circ\end{align*}. \begin{align*}m \angle D = 124^\circ\end{align*}.
Example B
Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Opposite sides are congruent, so we can set each pair equal to each other and solve both equations.
\begin{align*}6x-7&=2x+9 && y^2+3=12\\ 4x& =16 && \quad \ \ y^2=9\\ x&=4 && \quad \ \ \ y=3 \ or \ -3\end{align*}
Even though \begin{align*}y = 3\end{align*} or -3, lengths cannot be negative, so \begin{align*}y = 3\end{align*}.
Example C
Show that the diagonals of \begin{align*}FGHJ\end{align*} bisect each other.
The easiest way to show this is to find the midpoint of each diagonal. If it is the same point, you know they intersect at each other’s midpoint and, by definition, cuts a line in half.
\begin{align*}& \text{Midpoint of} \ \overline{FH}: \ \left ( \frac{-4+6}{2}, \ \frac{5-4}{2} \right ) = (1, 0.5)\\ & \text{Midpoint of} \ \overline{GJ} : \ \left ( \frac{3-1}{2}, \ \frac{3-2}{2} \right ) = (1, 0.5)\end{align*}
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter6ParallelogramsB
Concept Problem Revisited
By the Parallelogram Diagonals Theorem, the fountain is going to be 34 feet from either endpoint on the 68 foot diagonal and 25 feet from either endpoint on the 50 foot diagonal.
Vocabulary
A parallelogram is a quadrilateral with two pairs of parallel sides.
Guided Practice
1. \begin{align*}SAND\end{align*} is a parallelogram, \begin{align*}SY = 4x - 11\end{align*} and \begin{align*}YN = x + 10\end{align*}. Solve for \begin{align*}x\end{align*}.
2. Find the measures of \begin{align*}a\end{align*} and \begin{align*}b\end{align*} in the parallelogram below:
3. If \begin{align*}m \angle B = 72^\circ\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the other three angles.
Answers:
1. Because this is a parallelogram, the diagonals bisect each other and \begin{align*}SY\cong YN\end{align*}.
\begin{align*}SY & = YN\\ 4x - 11 & = x + 10\\ 3x & = 21\\ x & = 7\end{align*}
2. 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*}.
3. \begin{align*}m \angle D=72^\circ\end{align*} as well, because opposite angles are congruent. \begin{align*} \angle A\end{align*} and \begin{align*} \angle C\end{align*} are supplementary with \begin{align*} \angle D\end{align*}, so \begin{align*}m \angle A=m\angle C=108^\circ\end{align*}.
Interactive Practice
Practice
- If \begin{align*}m \angle S = 143^\circ\end{align*} in parallelogram \begin{align*}PQRS\end{align*}, find the other three angles.
- If \begin{align*}\overline{AB} \perp \overline{BC}\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the measure of all four angles.
- If \begin{align*}m \angle F = x^\circ\end{align*} in parallelogram \begin{align*}EFGH\end{align*}, find expressions for the other three angles in terms of \begin{align*}x\end{align*}.
For questions 4-11, find the measures of the variable(s). All the figures below are parallelograms.
Use the parallelogram \begin{align*}WAVE\end{align*} to find:
- \begin{align*}m \angle AWE\end{align*}
- \begin{align*}m \angle ESV\end{align*}
- \begin{align*}m \angle WEA\end{align*}
- \begin{align*}m \angle AVW\end{align*}
In the parallelogram \begin{align*}SNOW, ST = 6, NW = 4, m \angle OSW=36^\circ, m \angle SNW=58^\circ\end{align*} and \begin{align*}m \angle NTS=80^\circ\end{align*}. (diagram is not drawn to scale)
- \begin{align*}SO\end{align*}
- \begin{align*}NT\end{align*}
- \begin{align*}m \angle NWS\end{align*}
- \begin{align*}m \angle SOW\end{align*}
Plot the points \begin{align*}E(-1, 3), F(3, 4), G(5, -1), H(1, -2)\end{align*} and use parallelogram \begin{align*}EFGH\end{align*} for problems 20-23.
- Find the coordinates of the point at which the diagonals intersect. How did you do this?
- Find the slopes of all four sides. What do you notice?
- Use the distance formula to find the lengths of all four sides. What do you notice?
- Make a conjecture about how you might determine whether a quadrilateral in the coordinate is a parallelogram.
Write a two-column proof.
- 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*}
- 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*}
Use the diagram below to find the indicated lengths or angle measures for problems 26-29. The two quadrilaterals that share a side are parallelograms.
- \begin{align*}w\end{align*}
- \begin{align*}x\end{align*}
- \begin{align*}y\end{align*}
- \begin{align*}z\end{align*}