CK-12 Geometry
Page 23
How to identify and classify a square.
How to identify and classify a kite.
How to identify and classify a trapezoid.
How to identify and classify an isosceles trapezoid.
How to collect the classifications in a Venn diagram.
How to identify and classify shapes using a coordinate grid.
It is important to be able to classify different types of quadrilaterals in many different situations. The more you understand the differences and similarities between the shapes, the more success you’ll have applying them to more complicated problems.
Review Questions
Use the diagram below for exercises 4-7:
Find the slope of and , and find the slope of and .
Based on , what can you conclude now about quadrilateral ?
Find using the distance formula. What can you conclude about ?
If , find and .
Prove the Opposite Angles Theorem: The opposite angles of a parallelogram are congruent.
Draw a Venn diagram representing the relationship between Widgets, Wookies, and Wooblies (these are made-up terms) based on the following four statements: All Wookies are Wooblies
All Widgets are Wooblies
All Wookies are Widgets
Some Widgets are not Wookies
Sketch a kite. Describe the symmetry of the kite and write a sentence about what you know based on the symmetry of a kite.
Review Answers
The slope of and the slope of both since the lines are horizontal. For , . Finally for ,
Since the slopes of the opposite sides are equal, the opposite sides are parallel. Therefore, is a parallelogram
Using the distance formula,
Since is a parallelogram, we know that
and
First, we convert the theorem into “given” information and what we need to prove: Given: Parallelogram . Prove: and
Statement
Reason
1. is a parallelogram
1. Given
2. Draw auxiliary segment and label the angles as follows
2. Line Postulate
3.
3. Definition of parallelogram
4.
4. Alternate Interior Angles Theorem
5.
5. Definition of parallelogram
6.
6. Alternate Interior Angles Theorem
7.
7. Reflexive property
8.
8. ASA Triangle Congruence Postulate
9.
9. Definition of congruent triangles (all corresponding sides and angles of congruent triangles are congruent)
10.
10. Angle addition postulate
11.
11. Angle addition postulate
12.
12. Substitution
Now we have shown that opposite angles of a parallelogram are congruent
See below. The red line is a line of reflection. Given this symmetry, we can conclude that .
Using Parallelograms
Learning Objectives
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.
Introduction
Now that you have studied the different types of quadrilaterals and their defining characteristics, you can examine each one of them in greater depth. The first shape you’ll look at more closely is the parallelogram. It is defined as a quadrilateral with two pairs of parallel sides, but there are many more characteristics that make a parallelogram unique.
Opposite Sides in a Parallelogram
By now, you recognize that there are many types of parallelograms. They can look like squares, rectangles, or diamonds. Either way, opposite sides are always parallel. One of the most important things to know, however, is that opposite sides in a parallelogram are also congruent.
To test this theory, you can use pieces of string on your desk. Place two pieces of string that are the same length down so that they are parallel. You’ll notice that the only way to connect the remaining vertices will be two parallel, congruent segments. There will be only one possible fit given two lengths.
Try this again with two pieces of string that are different lengths. Again, lay them down so that they are parallel on your desk. What you should notice is that if the two segments are different lengths, the missing segments (if they connect the vertices) will not be parallel. Therefore, it will not create a parallelogram. In fact, there is no way to construct a parallelogram if opposite sides aren’t congruent.
So, even though parallelograms are defined by their parallel opposite sides, one of their properties is that opposite sides be congruent.
Example 1
Parallelogram is shown on the following coordinate grid. Use the distance formula to show that opposite sides in the parallelogram are congruent.
You can use the distance formula to find the length of each segment. You are trying to prove that is the same as , and that is the same as (Recall that means the same as , or the length of .)
Start with . The coordinates of are and the coordinates of are .
So .
Next find . The coordinates of are and the coordinates of are .
So .
Next find . The coordinates of are and the coordinates of are .
So .
Finally, find the length of . The coordinates of are and the coordinates of are .
So .
Thus, in parallelogram , and . The opposite sides are congruent.
This example shows that in this parallelogram, the opposite sides are congruent. In the last section we proved this fact is true for all parallelograms using congruent triangles. Here we have shown an example of this property in the coordinate plane.
Opposite Angles in a Parallelogram
Not only are opposite sides in a parallelogram congruent. Opposite angles are also congruent. You can prove this by drawing in a diagonal and showing ASA congruence between the two triangles created. Remember that when you have congruent triangles, all corresponding parts will be congruent.
Example 2
Fill in the blanks in the two-column proof below.
Given: is a parallelogram
Prove:
Statement Reason
1. is a parallelogram
1. Given
2.
2. Definition of a parallelogram
3.
3. Alternate Interior Angles Theorem
4.
4. Definition of a parallelogram
5.
5. _______________________
6.
6. Reflexive Property
7.
7. ASA Triangle Congruence Postulate
8.
8. Corresponding parts of congruent triangles are congruent
The missing statement in step 3 should be related to the information in step 2. and are parallel, and is a transversal. Look at the following figure (with the other segments removed) to see the angles formed by these segments:
Therefore the missing step is .
Work backwards to fill in step 4. Since step 5 is about , the sides we need parallel are and . So step 4 is .
The missing reason on step 5 will be the same as the missing reason in step 3: alternate interior angles.
Finally, to fill in the triangle congruence statement, BE CAREFUL to make sure you match up corresponding angles. The correct form is . (Students commonly get this reversed, so don’t feel bad if you take a few times to get it correct!)
As you can imagine, the same process could be repeated with diagonal to show that . Opposite angles in a parallelogram are congruent. Or, even better, we can use the fact that and together with the Angle Addition Postulate to show . We leave the details of these ope
rations for you to fill in.
Consecutive Angles in a Parallelogram
So at this point, you understand the relationships between opposite sides and opposite angles in parallelograms. 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 and . To help you understand the relationship, extend all of the segments involved with these angles and remove .
What you should notice is that and are two parallel lines cut by transversal . So, you can find the relationships between the angles as you learned in Chapter 1. Earlier in this course, you learned that in this scenario, two consecutive interior angles are supplementary; they sum to . The same is true within the parallelogram. Any two consecutive angles inside a parallelogram are supplementary.
Example 3
Fill in the remaining values for the angles in parallelogram below.
You already know that since it is given in the diagram. Since opposite angles are congruent, you can conclude that .
Now that you know that consecutive angles are supplementary, you can find the measures of the remaining angles by subtracting from .
So, . Since opposite angles are congruent, will also measure .
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. The easiest way to demonstrate this property is through congruent triangles, similarly to how we proved opposite angles congruent earlier in the lesson.
Example 4
Use a two-column proof for the theorem below.
Given: is a parallelogram
Prove: and
Statement Reason
1. is a parallelogram
1. Given.
2.
2. Opposite sides in a parallelogram are congruent.
3.
3. Vertical angles are congruent.
4.
4. Alternate interior angles are congruent.
5.
5. AAS congruence theorem: If two angles and one side in a triangle are congruent, the triangles are congruent.
6. and
6. Corresponding parts of congruent triangles are congruent.
Lesson Summary
In this lesson, we explored parallelograms. Specifically, we have learned:
How to describe and prove the distance relationships between opposite sides in a parallelogram.
How to describe and prove the relationship between opposite angles in a parallelogram.
How to describe and prove the relationship between consecutive angles in a parallelogram.
How to describe and prove the relationship between the two diagonals in a parallelogram.
It is helpful to be able to understand the unique properties of parallelograms. You will be able to use this information in many different ways.
Points to Consider
Now that you have learned the many relationships in parallelograms, it is time to learn how you can prove that shapes are parallelograms.
Review Questions
Use the following figure for exercises 3-6.
Find the slopes of and .
Find the slopes of and .
What kind of quadrilateral is ? Give an answer that is as detailed as possible.
If you add diagonals to , where will they intersect?
Use the figure below for questions 7-11. Polygon is a regular polygon. Find each indicated measurement.
What kind of triangle is ?
Copy polygon and add auxiliary lines to make each of the following: a parallelogram
a trapezoid
an isosceles triangle
Review Answers
Slopes of and
This figure is a parallelogram since opposite sides have equal slopes (i.e., opposite sides are parallel). Additionally, it is a rectangle because each angle is a angle. We know this because the slopes of adjacent sides are opposite reciprocals
The diagonals would intersect at . One way to see this is to use the symmetry of the figure—each corner is a rotation around the origin from adjacent corners
is an isosceles right triangle
There are many possible answers. Here is one: Auxiliary lines are in red:
is a parallelogram (in fact it is a rectangle).
is a trapezoid.
is an isosceles triangle
Proving Quadrilaterals are Parallelograms
Learning Objectives
Prove a quadrilateral is a parallelogram given congruent opposite sides.
Prove a quadrilateral is a parallelogram given congruent opposite angles.
Prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
Prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.
Introduction
You’ll remember from earlier in this course that you have studied converse statements. A converse statement reverses the order of the hypothesis and conclusion in an if-then statement, and is only sometimes true. For example, consider the statement: “If you study hard, then you will get good grades.” Hopefully this is true! However, the converse is “If you get good grades, then you study hard.” This may be true, but is it not necessarily true—maybe there are many other reasons why you get good grades—i.e., the class is really easy!
An example of a statement that is true and whose converse is also true is as follows: If I face east and then turn a quarter-turn to the right, I am facing south. Similarly, if I turn a quarter-turn to the right and I am facing south, then I was facing east to begin with.
Also all geometric definitions have true converses. For example, if a polygon is a quadrilateral then it has four sides and if a polygon has four sides then it is a quadrilateral.
Converse statements are important in geometry. It is crucial to know which theorems have true converses. In the case of parallelograms, almost all of the theorems you have studied this far have true converses. This lesson explores which characteristics of quadrilaterals ensure that they are parallelograms.
Proving a Quadrilateral is a Parallelogram Given Congruent Sides
In the last lesson, you learned that a parallelogram has congruent opposite sides. We proved this earlier and then looked at one example of this using the distance formula on a coordinate grid to verify that opposite sides of a parallelogram had identical lengths.
Here, we will show on the coordinate grid that the converse of this statement is also true: If a quadrilateral has two pairs of opposite sides that are congruent, then it is a parallelogram.
Example 1
Show that the figure on the grid below is a parallelogram.
We can see that the lengths of opposite sides in this quadrilateral are congruent. For example, to find the length of we can find the difference in the coordinates because is horizontal (it’s generally very easy to find the length of horizontal and vertical segments). and . So, we have established that opposite sides of this quadrilateral are congruent.
But is it a parallelogram? Yes. One way to argue that is a parallelogram is to note that . We can think of as a transversal that crosses and . Now, interior angles on the same side of the transversal are supplementary, so we can apply the postulate if interior angles on the same side of the transversal are supplementary then the lines crossed by the transversal are parallel.
Note: This example does not prove that if opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram. To do that you need to use any quadrilateral with congruent opposite sides, and then you use congruent triangles to help you. We will let you do that as an exercise, but here’s the basic picture. What triangle congruence postulate can you use to show
Proving a Quadrilateral is a Parallelogram Given Congruent Opposite Angles
Much like the converse statements you studied about opposite side lengths, if you can prove that opposite angles in a quadrilateral are congruent, the figure is a parallelogram.
Example 2
Complete the two-column proof below.
Given: Quadrilateral with and
Prove: is a parallelogram
Statement
Reason
1. is a quadrilateral with and
1. Given
2.
2. Sum of the angles in a quadrilateral is
3.
3. Substitution and
4.
4. Combine like terms
5.
5. Factoring
6.
6. Division property of equality (divided both sides by 2)
7.
7. If interior angles on the same side of a transversal are supplementary then the lines crossed by the transversal are parallel
8.
8. Substitution on line 6
9.
9. Same reason as step 7
10. is a parallelogram 10. Definition of a parallelogram