2.8: Properties of Parallelograms
Learning Objectives
- Identify properties of parallelograms.
- Describe the relationships between opposite sides in a parallelogram.
- Describe the relationship between opposite angles in a parallelogram.
- Describe the relationship between consecutive angles in a parallelogram.
- Describe the relationship between the two diagonals in a parallelogram.
- Apply parallelogram properties to solve problems.
Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides. Each of the shapes shown below is a parallelogram:
Do you remember what a quadrilateral is?
It is a polygon with four sides.
Some other words that have the same prefix as quadrilateral are:
- quadruple – to multiply by 4
- quarter – one fourth
- quadruplets – four brothers and sisters born at the same time
As you can see, parallelograms come in a variety of shapes. The only defining feature is that opposite sides are parallel. But, once we know that a figure is a parallelogram, we have very useful theorems we can use to solve problems involving parallelograms. A parallelogram is a _______________________________ with two pairs of parallel sides.
A quadrilateral is a polygon with ____________ sides.
Reading Check:
1. What makes a quadrilateral a parallelogram?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Mark the parallel lines with arrows ( > and >> ) to show the pairs of parallel lines in the picture below:
Opposite Sides in a Parallelogram
There are many types of parallelograms. Opposite sides are always parallel.
One of the most important things to know, however, is that opposite sides in a parallelogram are also congruent.
Opposite Sides of Parallelogram Theorem
The opposite sides of a parallelogram are congruent.
Look at the pair of parallel line segments here:
__________________________
__________________________
If you want to connect the endpoints of the line segments, you must draw in two parallel, congruent line segments.
If the line segments are not congruent, you cannot make a parallelogram:
So, even though parallelograms are defined by their parallel opposite sides, one of their properties is that opposite sides be congruent.
Opposite Angles in a Parallelogram
Not only are opposite sides in a parallelogram congruent; opposite angles are also congruent.
Opposite Angles in Parallelogram Theorem
The opposite angles of a parallelogram are congruent.
You have learned that when lines are parallel, their corresponding angles are congruent.
Opposite sides and opposite ____________________ in a parallelogram are congruent.
Reading Check:
Mark as many measurements as you can in the picture of the parallelogram below.
Consecutive Angles in a Parallelogram
At this point, you understand the relationships between opposite sides and opposite angles in parallelograms.
In a parallelogram...
- Opposite sides are parallel (this is the definition of a parallelogram)
- Opposite sides are congruent
- Opposite angles are congruent
Think about the relationship between consecutive angles in a parallelogram. You have studied this scenario before, but you can apply what you have learned to parallelograms.
Examine the parallelogram below:
Imagine that you are trying to find the relationship between \begin{align*}\angle SPQ\end{align*} and \begin{align*}\angle PSR\end{align*} in the diagram on the previous page. To help you understand the relationship, extend all of the segments involved with these angles and remove \begin{align*}\overline{RQ}\end{align*} like we have below:
What you should notice is that \begin{align*}\overleftrightarrow{PQ}\end{align*} and \begin{align*}\overleftrightarrow{SR}\end{align*} are two parallel lines cut by transversal \begin{align*}\overleftrightarrow{SP}\end{align*}.
- \begin{align*}\overleftrightarrow{PQ}\end{align*} and \begin{align*}\overleftrightarrow{SR}\end{align*} are ______________________________ lines.
- \begin{align*}\angle SPQ\end{align*} and \begin{align*}\angle PSR\end{align*} are same-side ______________________________ angles.
Earlier in this lesson, you learned that in this scenario, two consecutive interior angles are supplementary; they sum to \begin{align*}180^\circ\end{align*}. The same is true within the parallelogram. Any two consecutive angles inside a parallelogram are supplementary.
Consecutive Angles in Parallelogram Theorem
Any two consecutive angles of a parallelogram are supplementary.
Diagonals in a Parallelogram
There is one more relationship to examine within parallelograms. When you draw the two diagonals inside parallelograms, they bisect each other. This can be very useful information for examining larger shapes that may include parallelograms.
Diagonals in a Parallelogram Theorem
The diagonals of a parallelogram bisect one another.
Bisecting diagonals means that the ____________________________ cut each other in half.
Reading Check:
1. True or False: Opposite angles in a parallelogram are congruent.
2. True or False: Consecutive angles in a parallelogram are congruent.
3. True or False: Opposite sides in a parallelogram are both congruent and parallel.
4. Find the measure of the missing segments in the picture below.
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