6.8: Understanding the Angle Measures of Quadrilaterals
Have you ever used a quadrilateral in a realworld object? Take a look at this dilemma.
Margie makes jewelry. She made this necklace to sell at a craft fair.
Can you identify the quadrilateral? In this Concept, you will learn how to accomplish this task.
Guidance
What is a quadrilateral?
A quadrilateral is any foursided figure.
In the word “quadrilateral”, we find the word “quad” which means four. This means that any foursided figure is considered a quadrilateral. Now, there are different types of quadrilaterals that we are going to learn about in this lesson.
We can say that a quadrilateral is any foursided figure. We could consider this an umbrella category meaning that there are different types of quadrilaterals that we can identify in a specific way even though they are still quadrilaterals too.
Let’s look at identify the types of quadrilaterals.
The first type of quadrilateral to learn about is called a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel and congruent.
Here is a picture of a parallelogram.
When you look at this picture, you can see that the opposite sides of the figure are parallel. They are also the same lengthmeaning congruent.
There are three main kinds of parallelograms.
Parallelograms can be plain old parallelograms like the one in the picture. They could also be a rectangle, square and rhombus.
A rectangle is a parallelogram with four right angles, where opposite sides are congruent and parallel. You have been looking at rectangles for a long time, but now you need to notice that there are specific properties that make a rectangle a rectangle.
A rhombus is a parallelogram with four congruent sides, but not necessarily four right angles. A rhombus can look like a square, but while a square is always a rhombus, a rhombus is not necessarily a square. A rhombus can only be a square if it has four right angles.
A square is a parallelogram too. The big difference between a square and a rectangle is that a square has four congruent sides. It also has four right angles though just like a rectangle.
There is one other type of quadrilateral. This quadrilateral is NOT a parallelogram. It is a special kind of quadrilateral. It is called a trapezoid. A trapezoid is a quadrilateral with one pair of opposite sides parallel.
One important thing to remember about quadrilaterals is that their four angles always have a sum of \begin{align*}360^\circ\end{align*}
Notice how different the angles and the sides of the quadrilaterals are. Look closely, though. If you add up the measures of the four angles, they always equal \begin{align*}360^\circ\end{align*}
This quadrilateral has been divided into two congruent triangles, each with angles of \begin{align*}120^\circ, 25^\circ\end{align*}
We can use what we know about quadrilaterals to analyze them. When we analyze quadrilaterals, we can find the measure of an unknown angle or side. Remember, one of the most important things to know about quadrilaterals is that their angles always add up to \begin{align*}360^\circ\end{align*}
Find the measure of the unknown angle in the quadrilateral below.
We know that the four angles must have a sum of \begin{align*}360^\circ\end{align*}
\begin{align*}55+90+105+m &= 360\\
250+m &= 360\\
m &= 360250\\
m &= 110^\circ\end{align*}
By solving for \begin{align*}m\end{align*}
We can check our work by adding the four angles to see if they total \begin{align*}360^\circ\end{align*}
\begin{align*}55^\circ + 90^\circ + 105^\circ + 110^\circ = 360^\circ\end{align*}
Our calculation was correct. We can always use this method when given three out of the four angles in a quadrilateral.
Often we can use what we know about the properties of quadrilaterals to find unknown measures without having to set up an equation. We can simply use reasoning to put the pieces together.
Identify each missing angle.
Example A
\begin{align*}110^\circ, 110^\circ, 70^\circ,?\end{align*}
Solution:\begin{align*}70^\circ\end{align*}
Example B
\begin{align*}90^\circ, 90^\circ, 90^\circ,?\end{align*}
Solution:\begin{align*}90^\circ\end{align*}
Example C
\begin{align*}100^\circ, 100^\circ, 80^\circ,?\end{align*}
Solution:\begin{align*}80^\circ\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
Look at the necklace that Margie made once again.
Now let’s examine this picture. We can look for the qualities that identify this quadrilateral. Notice that it has two parallel sides. The other two sides aren’t parallel or congruent. With one pair of parallel sides, this figure must be a trapezoid.
Vocabulary
 Quadrilateral
 any foursided figure.
 Trapezoid
 a quadrilateral with one pair of parallel sides.
 Parallelogram
 a quadrilateral with two pairs of opposite sides that are congruent and parallel.
 Rhombus
 a parallelogram with four congruent sides.
 Rectangle
 a parallelogram with opposites congruent and four right angles.
 Square
 a parallelogram with four congruent sides and four right angles.
 Congruent
 means exactly the same.
Guided Practice
Here is one for you to try on your own.
Find the measures of the unknown angles in the quadrilateral below.
Solution
This time we have only been given the measures of two angles and we need to solve for the other two. First let’s determine what we know about the figure. What kind of quadrilateral is it? It has two pairs of parallel sides, so it must be a parallelogram. It doesn’t have \begin{align*}90^\circ\end{align*}
Now, what do we know about the angles of parallelograms? Not only do they add up to \begin{align*}360^\circ\end{align*}
Angle \begin{align*}x\end{align*}
Let’s check to make sure these are the correct measurements by adding them to see if they total \begin{align*}360^\circ\end{align*}
\begin{align*}124^\circ + 124^\circ + 56^\circ + 56^\circ = 360^\circ\end{align*}
They do, so our answers are correct.
Video Review
Khan Academy Overview of Quadrilaterals
Practice
Directions: Use what you have learned about quadrilaterals to figure out the missing angle measure of each quadrilateral based on three given angles.

\begin{align*}120^\circ, 120^\circ, 60^\circ,?\end{align*}
120∘,120∘,60∘,? 
\begin{align*}50^\circ, 70^\circ, 130^\circ,?\end{align*}
50∘,70∘,130∘,? 
\begin{align*}52^\circ, 128^\circ, 52^\circ,?\end{align*}
52∘,128∘,52∘,? 
\begin{align*}47^\circ, 55^\circ, 120^\circ,?\end{align*}
47∘,55∘,120∘,? 
\begin{align*}80^\circ, 80^\circ, 100^\circ,?\end{align*}
80∘,80∘,100∘,? 
\begin{align*}105^\circ, 105^\circ, 85^\circ,?\end{align*}
105∘,105∘,85∘,? 
\begin{align*}97^\circ, 97^\circ, 35^\circ,?\end{align*}
97∘,97∘,35∘,? 
\begin{align*}120^\circ, 120^\circ, 40^\circ,?\end{align*}
120∘,120∘,40∘,? 
\begin{align*}88^\circ, 90^\circ, 60^\circ,?\end{align*}
88∘,90∘,60∘,? 
\begin{align*}25^\circ, 85^\circ, 85^\circ,?\end{align*}
25∘,85∘,85∘,? 
\begin{align*}90^\circ, 90^\circ, 90^\circ,?\end{align*}
90∘,90∘,90∘,?  \begin{align*}140^\circ, 150^\circ, 45^\circ,?\end{align*}
 \begin{align*}80^\circ, 80^\circ, 120^\circ,?\end{align*}
 \begin{align*}75^\circ, 95^\circ, 110^\circ,?\end{align*}
 \begin{align*}80^\circ, 50^\circ, 95^\circ,?\end{align*}
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Congruent
Congruent figures are identical in size, shape and measure.Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides.Quadrilateral
A quadrilateral is a closed figure with four sides and four vertices.Rectangle
A rectangle is a quadrilateral with four right angles.Rhombus
A rhombus is a quadrilateral with four congruent sides.Square
A square is a polygon with four congruent sides and four right angles.Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.Image Attributions
Here you'll understand the angle measures of quadrilaterals.