Proving a Quadrilateral is a Parallelogram Given Bisecting Diagonals
In the last lesson, you learned that in a parallelogram, the diagonals bisect each other. This can also be turned around into a converse statement. If you have a quadrilateral in which the diagonals bisect each other, then the figure is a parallelogram. See if you can follow the proof below which shows how this is explained.
Example 3
Complete the two-column proof below.
Given: , and
Prove: is a parallelogram
Statement
Reason
1.
1. Given
2.
2. Given
3.
3. Vertical angles are congruent
4.
4.
If two sides and the angle between them are congruent, the two triangles are congruent
5.
5. Corresponding parts of congruent triangles are congruent
6.
6. Vertical angles are congruent
7.
7.
If two sides and the angle between them are congruent, then the two triangles are congruent
8.
8. Corresponding parts of congruent triangle are congruent
9. is a parallelogram
9. If two pairs of opposite sides of a quadrilateral are congruent, the figure is a parallelogram
So, given only the information that the diagonals bisect each other, you can prove that the shape is a parallelogram.
Proving a Quadrilateral is a Parallelogram Given One Pair of Congruent and Parallel Sides
The last way you can prove a shape is a parallelogram involves only one pair of sides.
The proof is very similar to the previous proofs you have done in this section so we will leave it as an exercise for you to fill in. To set up the proof (which often IS the most difficult step), draw the following:
Given: Quadrilateral with and
Prove: is a parallelogram
Example 4
Examine the quadrilateral on the coordinate grid below. Can you show that it is a parallelogram?
To show that this shape is a parallelogram, you could find all of the lengths and compare opposite sides. However, you can also study one pair of sides. If they are both congruent and parallel, then the shape is a parallelogram.
Begin by showing two sides are congruent. You can use the distance formula to do this.
Find the length of . Use for and for .
Next, find the length of the opposite side, . Use for and for .
So, ; they have equal lengths. Now you need to show that and are parallel. You can do this by finding their slopes. Recall that if two lines have the same slope, they are parallel.
So, the slope of . Now, check the slope of .
So, the slope of . Since the slopes of and are the same, the two segments are parallel. Now that have shown that the opposite segments are both parallel and congruent, you can identify that the shape is a parallelogram.
Lesson Summary
In this lesson, we explored parallelograms. Specifically, we have learned:
How to prove a quadrilateral is a parallelogram given congruent opposite sides.
How to prove a quadrilateral is a parallelogram given congruent opposite angles.
How to prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
How to prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.
It is helpful to be able to prove that certain quadrilaterals are parallelograms. You will be able to use this information in many different ways.
Review Questions
Use the following diagram for exercises 1-3.
Find each angle:
If and , find each length:
If and , find each length:
Use the following figure for exercises 4-7.
Suppose that , , and are three of four vertices (corners) of a parallelogram. Give two possible locations for the fourth vertex, , if you know that the coordinate of is .
Depending on where you choose to put point in , the name of the parallelogram you draw will change. Sketch a picture to show why.
If you know the parallelogram is named , what is the slope of the side parallel to ?
Again, assuming the parallelogram is named , what is the length of ?
Prove: If opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Given: with and
Prove: and (i.e., is a parallelogram).
Prove: If a quadrilateral has one pair of congruent parallel sides, then it is a parallelogram.
Note in 9 that the parallel sides must also be the congruent sides for that theorem to work. Sketch a counterexample to show that if a quadrilateral has one pair of parallel sides and one pair of congruent sides (which are not the parallel sides) then the resulting figure it is not necessarily a parallelogram. What kind of quadrilaterals can you make with this arrangement?
Review Answers
(note: you need to find almost all angle measures in the diagram to answer this question)
,
can be at either or
If is at the parallelogram would be named (in red in the following illustration). If is at then the parallelogram will take the name .
would have a slope of .
Given: with and Prove: and (i.e., is a parallelogram)
Statement
Reason
1.
1. Given
2.
2. Given
3. Add auxiliary line
3. Line Postulate
4.
4. Reflexive Property
5.
5. SSS Congruence Postulate
6.
6. Definition of congruent triangles
7.
7. Converse of Alternate Interior Angles Postulate
8.
8. Definition of congruent triangles
9.
9. Converse of Alternate Interior Angles Postulate
First, translate the theorem into given and prove statements: Given: with and
Prove:
Statement
Reason
1.
1. Given
2.
2. Given
3.
3. Alternate Interior Angles Theorem
4. Add auxiliary line
4. Line Postulate
5.
5. Reflexive Property
6.
6. SAS Triangle Congruence Postulate
7.
7. Definition of congruent triangles
8.
8. Converse of Alternate Interior Angles Theorem
If the congruent sides are not the parallel sides, then you can make either a parallelogram (in black) or an isosceles trapezoid (in red):
Rhombi, Rectangles, and Squares
Learning Objectives
Identify the relationship between the diagonals in a rectangle.
Identify the relationship between diagonals in a rhombus.
Identify the relationship between diagonals and opposite angles in a rhombus.
Identify and explain biconditional statements.
Introduction
Now that you have a much better understanding of parallelograms, you can begin to look more carefully into certain types of parallelograms. This lesson explores two very important types of parallelograms—rectangles and rhombi. Remember that all of the rules that apply to parallelograms still apply to rectangles and rhombi. In this lesson, you’ll learn about rules specific to these shapes that are not true for all parallelograms.
Diagonals in a Rectangle
Recall from previous lessons that the diagonals in a parallelogram bisect each other. You can prove this with congruence of triangles within the parallelogram. In a rectangle, there is an even more special relationship between the diagonals. The two diagonals in a rectangle will always be congruent
. We can show this using the distance formula on a coordinate grid.
Example 1
Use the distance formula to demonstrate that the two diagonals in the rectangle below are congruent.
To solve this problem, you need to find the lengths of both diagonals in the rectangle. First, draw line segments that connect the vertices of the rectangle. So, draw a segment from to and from to .
You can use the distance formula to find the length of the diagonals. Diagonal goes from to .
Next, find the length of diagonal . That diagonal goes from to .
So, . In this example, the diagonals are congruent. Are the diagonals of rectangles always congruent? The answer is yes.
Theorem: The diagonals of a rectangle are congruent
The proof of this theorem relies on the definition of a rectangle (a quadrilateral in which all angles are congruent) as well as the property that rectangles are parallelograms.
Given: Rectangle
Prove:
Statement
Reason
1. is a rectangle
1. Given
2.
2. Definition of a rectangle
3.
3. Opposite sides of a parallelogram are
4.
4. Reflexive Property of
5.
5. SAS Congruence Postulate
6.
6. Definition of congruent triangles (corresponding parts of congruent triangles are congruent)
Perpendicular Diagonals in Rhombi
Remember that rhombi are quadrilaterals that have four congruent sides. They don’t necessarily have right angles (like squares), but they are also parallelograms. Also, all squares are parallelograms.
The diagonals of a rhombus not only bisect each other (because they are parallelograms), they do so at a right angle. In other words, the diagonals are perpendicular. This can be very helpful when you need to measure angles inside rhombi or squares.
Theorem: The diagonals of a rhombus are perpendicular bisectors of each other
The proof of this theorem uses the fact that the diagonals of a parallelogram bisect each other and that if two angles are congruent and supplementary, then they are right angles.
Given: Rhombus with diagonals and intersecting at point A
Prove:
Statement
Reason
1. is a rhombus
1. Given
2. is a parallelogram
2. Theorem: All rhombi are parallelograms
3.
3. Definition of a rhombus
4.
4. Reflexive Property of
5.
5. Diagonals of a parallelogram bisect each other
6.
6. Triangle Congruence Postulate
7.
7. Definition of congruent triangles (corresponding parts of congruent triangles are congruent)
8. and are supplementary
8. Linear Pair Postulate
9. and are right angles
9. Congruent supplementary angles are right angles
10.
10. Definition of perpendicular lines
Remember that you can also show that lines or segments are perpendicular by comparing their slopes. Perpendicular lines have slopes that are opposite reciprocals of each other.
Example 2
Analyze the slope of the diagonals in the rhombus below. Use slope to demonstrate that they are perpendicular.
Notice that the diagonals in this diagram have already been drawn in for you. To find the slope, find the change in over the change in . This is also referred to as rise over run.
Begin by finding the slope of the diagonal , which goes from to.
Now find the slope of the diagonal from to .
The slope of and the slope of . These two slopes are opposite reciprocals of each other, so the two segments are perpendicular.
Diagonals as Angle Bisectors
Since a rhombus is a parallelogram, opposite angles are congruent. One property unique to rhombi is that in any rhombus, the diagonals will bisect the interior angles. Here we will prove this theorem using a different method than the proof we showed above.
Theorem: The diagonals of a rhombus bisect the interior angles
Example 3
Complete the two-column proof below.
Given: is a rhombus
Prove:
Statement
Reason
1. is a rhombus
1. Given
2.
2. All sides in a rhombus are congruent
3. is isosceles
3. Any triangle with two congruent sides is isosceles
4.
4. The base angles in an isosceles triangle are congruent
5.
5. Alternate interior angles are congruent
6.
6. Transitive Property
Segment bisects . You could write a similar proof for every angle in the rhombus. Diagonals in rhombi bisect the interior angles.
Biconditional Statements
Recall that a conditional statement is a statement in the form “If then .” For example, if a quadrilateral is a parallelogram, then opposite sides are congruent.
You have learned a number of theorems as conditional statements. Many times you have also investigated the converses of these theorems. Sometimes the converse of a statement is true, and sometimes the converse are not. For example, you could say that if you live in Los Angeles, you live in California. However, the converse of this statement is not true. If you live in California, you don’t necessarily live in Los Angeles.
A biconditional statement is a conditional statement that also has a true converse. For example, a true biconditional statement is, “If a quadrilateral is a square then it has exactly four congruent sides and four congruent angles.” This statement is true, as is its converse: “If a quadrilateral has exactly four congruent sides and four congruent angles, then that quadrilateral is a square.” When a conditional statement can be written as a biconditional, then we use the term “if and only if.” In the previous example, we could say: “A quadrilateral is a square if and only if it has four congruent sides and four congruent angles.”
Example 4
Which of the following is a true biconditional statement?
A. A polygon is a square if and only if it has four right angles.
B. A polygon is a rhombus if and only if its diagonals are perpendicular bisectors.
C. A polygon is a parallelogram if and only if its diagonals bisect the interior angles.
D. A polygon is a rectangle if and only if its diagonals bisect each other.
Examine each of the statements to see if it is true. Begin with choice A. It is true that if a polygon is a square, it has four right angles. However, the converse statement is not necessarily true. A rectangle also has four right angles, and a rectangle is not necessarily a square. Providing an example that shows something is not true is called a counterexample.
The second statement seems correct. It is true that rhombi have diagonals that are perpendicular bisectors. The same is also true in converse—if a figure has perpendicular bisectors as diagonals, it is a rhombus. Check the other statements to make sure that they are not biconditionally true.
The third statement isn’t necessarily true. While rhombi have diagonals that bisect the interior angles, it is not true of all parallelograms. Choice C is not biconditionally true.
The fourth statement is also not necessarily true. The diagonals in a rectangle do bisect each other, but parallelograms that are not rectangles also have bisecting diagonals. Choice D is not correct.
So, after analyzing each statement carefully, only B is true. Choice B is the correct answer.
Lesson Summary
In this lesson, we explored rhombi, rectangles, and squares. Specifically, we have learned:
How to identify and prove the relationship between the diagonals in a rectangle.
How to identify and pro
ve the relationship between diagonals in a rhombus.
How to identify and prove the relationship between diagonals and opposite angles in a rhombus.
How to identify and explain biconditional statements.
It is helpful to be able to identify specific properties in quadrilaterals. You will be able to use this information in many different ways.
Review Questions
Use Rectangle for exercises 1-3.
Use rhombus for exercises 4-7.
If . and ., then _____
_____
What is the perimeter of ?
is the ______________________ of
For exercises 8 and 9, rewrite each given statement as a biconditional statement. Then state whether it is true. If the statement is false, draw a counterexample.
If a quadrilateral is a square, then it is a rhombus.
If a quadrilateral has for right angles, then it is a rectangle.
Give an example of an if-then statement whose converse is true. Then write that statement as a biconditional.
CK-12 Geometry Page 24