Download Understanding the Angle Measures of Quadrilaterals

Understanding the Angle
Measures of Quadrilaterals
Jen Kershaw
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Printed: October 28, 2013
AUTHOR
Jen Kershaw
www.ck12.org
C ONCEPT
Concept 1. Understanding the Angle Measures of Quadrilaterals
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Understanding the Angle
Measures of Quadrilaterals
Here you’ll understand the angle measures of quadrilaterals.
Have you ever used a quadrilateral in a real-world 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 four-sided figure.
In the word “quadrilateral”, we find the word “quad” which means four. This means that any four-sided 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 four-sided 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.
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When you look at this picture, you can see that the opposite sides of the figure are parallel. They are also the same
length-meaning 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.
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Concept 1. Understanding the Angle Measures of Quadrilaterals
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.
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One important thing to remember about quadrilaterals is that their four angles always have a sum of 360◦ .
This is true no matter what shape or size the quadrilateral is.
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 360◦ . This is because every quadrilateral is actually two triangles
put together. As we know, the three angles in all triangles always add up to 180◦ .
This quadrilateral has been divided into two congruent triangles, each with angles of 120◦ , 25◦ , and 35◦ . If we add
these angles together, we get a sum of 180◦ . If we step back and look at the whole quadrilateral, we see that it
has two 120◦ angles and two 60◦ angles (25◦ + 35◦ = 60◦ ). When we add these together, we get a sum of 360◦ :
60◦ + 120◦ + 60 + 120◦ = 360◦ . This will be true no matter what size each angle in the quadrilateral measures.
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 360◦ . That means that if we know the measure of any three angles, we can set up
an equation to solve for the measure of the fourth. Let’s see how this works.
Find the measure of the unknown angle in the quadrilateral below.
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Concept 1. Understanding the Angle Measures of Quadrilaterals
We know that the four angles must have a sum of 360◦ , so we can add the four angles, using m to represent
the unknown angle.
55 + 90 + 105 + m = 360
250 + m = 360
m = 360 − 250
m = 110◦
By solving for m, we have found that the fourth angle has a measure of 110◦ .
We can check our work by adding the four angles to see if they total 360◦ .
55◦ + 90◦ + 105◦ + 110◦ = 360◦
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
110◦ , 110◦ , 70◦ , ?
Solution: 70◦
Example B
90◦ , 90◦ , 90◦ , ?
Solution: 90◦
Example C
100◦ , 100◦ , 80◦ , ?
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Solution: 80◦
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 four-sided 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.
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Concept 1. Understanding the Angle Measures of Quadrilaterals
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 90◦ angles, so it’s not a rectangle or square. Finally, the side lengths are not
all congruent so it cannot be a rhombus. It is a regular parallelogram.
Now, what do we know about the angles of parallelograms? Not only do they add up to 360◦ , they fall into
two congruent pairs. The congruent angles are opposite each other. Take a look back at the figure.
Angle x is opposite the 56◦ angle. Therefore it must also be 56◦ . Angle y is opposite the 124◦ angle, so it must
also be 124◦ . This gives us two pairs of congruent angles.
Let’s check to make sure these are the correct measurements by adding them to see if they total 360◦ .
124◦ + 124◦ + 56◦ + 56◦ = 360◦
They do, so our answers are correct.
Video Review
MEDIA
Click image to the left for more content.
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.
1. 120◦ , 120◦ , 60◦ , ?
2. 50◦ , 70◦ , 130◦ , ?
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3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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52◦ , 128◦ , 52◦ , ?
47◦ , 55◦ , 120◦ , ?
80◦ , 80◦ , 100◦ , ?
105◦ , 105◦ , 85◦ , ?
97◦ , 97◦ , 35◦ , ?
120◦ , 120◦ , 40◦ , ?
88◦ , 90◦ , 60◦ , ?
25◦ , 85◦ , 85◦ , ?
90◦ , 90◦ , 90◦ , ?
140◦ , 150◦ , 45◦ , ?
80◦ , 80◦ , 120◦ , ?
75◦ , 95◦ , 110◦ , ?
80◦ , 50◦ , 95◦ , ?