6 4 homework proving a quadrilateral is a parallelogram

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High school geometry

Course: high school geometry   >   unit 3.

• Proof: Opposite sides of a parallelogram
• Proof: Diagonals of a parallelogram
• Proof: Opposite angles of a parallelogram
• Proof: The diagonals of a kite are perpendicular
• Proof: Rhombus diagonals are perpendicular bisectors
• Proof: Rhombus area

Prove parallelogram properties

• (Choice A)   A C ― ≅ B D ― ‍   A A C ― ≅ B D ― ‍  
• (Choice B)   △ A B C ≅ △ C D A ‍   B △ A B C ≅ △ C D A ‍  
• (Choice C)   A B ― ≅ D C ― ‍   C A B ― ≅ D C ― ‍  
• (Choice D)   ∠ A B D ≅ ∠ D B C ‍   D ∠ A B D ≅ ∠ D B C ‍  

Proving a Quadrilateral is a Parallelogram

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Lesson Vocabulary

Use properties of sides, angles, and diagonals to identify a parallelogram., essential question, which properties determine whether a quadrilateral is a parallelogram.

Proving Quadrilaterals Are Parallelograms

In the previous section, we learned about several properties that distinguish parallelograms from other quadrilaterals . Most of the work we did was computation-based because we were already given the fact that the figures were parallelograms. In this section, we will use our reasoning skills to put together two-column geometric proofs for parallelograms. We can apply much of what we learned in the previous section to help us throughout this lesson, but we will be much more formalized and organized in our arguments.

Using Definitions and Theorems in Proofs

The ways we start off our proofs are key steps toward arriving at a conclusion. Therefore, comprehending the information that we are given by an exercise may be the single most important part of proving a statement.

As we will see, there are different ways in which we can essentially say the same statement. Recall, that many of our angle theorems had converses. The converses of the theorems essentially gave the same information, but in a reversed order. We will have to approach problems involving parallelograms in the same way. That is, we must be conscious of the arguments we make based on whether we are given that a certain quadrilateral is a parallelogram, or if we want to prove that the quadrilateral is a parallelogram. Let’s take a look at these statements so that we understand how to use them properly in our proofs.

Given a Parallelogram

We can use the following statements in our proofs if we are given that a quadrilateral is a parallelogram.

Definition: A parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.

If a quadrilateral is a parallelogram, then…

Much of the information above was studied in the previous section. The purpose of organizing it in the way that it has been laid out is to help us see the difference in our statements depending on whether we are given a parallelogram, or if we are trying to prove that a quadrilateral is a parallelogram.

Let’s look at the structure of our statements when we are trying to prove that a quadrilateral is a parallelogram.

Proving a Parallelogram

…the quadrilateral is a parallelogram.

Let’s use these statements to help us prove the following exercise. We will need to use both forms of the statements above, because we will be given one parallelogram, and we will have to prove that another one exists. This will give us practice using regular theorems and definitions, as well as their converses.

As stated before this exercise, we need to be conscious of how to use theorems and definitions, as well as their converses because we are given that NRSM is a parallelogram, but we also want to prove that ERAM is a parallelogram. We were also given that ?4??5 , which will help us prove our conclusion.

To begin, we know that ?R??M because they are the opposite angles of parallelogram NRSM .

Knowing this allows us to claim that ?3??6 by the Angle Subtraction Postulate . We see that ?R is composed of two smaller angles ( ?3 and ?4 ). Likewise, we see that ?M is composed of ?5 and ?6 . Since the whole of the angles are congruent, and two of the smaller angles in them are congruent, then their remainders are also congruent.

Now, we have proven that one pair of opposite angles are congruent. If we can show that ?2 and ?7 are also congruent, we can prove that quadrilateral ERAM is a parallelogram.

Because NRSM is a parallelogram, we know that its opposite sides are parallel. So, we have that segments NR and MS are parallel. Considering these lines, we know that segments EM and RA are transversals to the parallel lines, since they intersect both lines. Thus, we can use the Alternate Interior Angles Theorem to prove that ?1??6 and ?3??8 .

By transitivity , we can say that ?1 is congruent to ?8 . It is a bit difficult to imagine the chain of congruences that allows us to make this claim, but it is as such:

Notice that ?3 and ?6 are congruent, opposite angles, just as ?2 and ?7 are. Let’s look at our new illustration to help us visualize what we’ve done.

We have proven that ERAM is a parallelogram because both pairs of its opposite angles are congruent. The two-column proof for our argument is shown below.

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Prove a quadrilateral is a parallelogram

Criteria needed to prove a shape is a parallogram.

There are 5 distinct ways to know that a quadrilateral is a paralleogram. If a quadrilateral meets any of the 5 criteria below, then it must be a parallelogram.

Criteria proving a quadrilateral is parallelogram

• 1) If a quadrilateral has one pair of sides that are both parallel and congruent .
• 2) If all opposite sides of the quadrilateral are congruent .
• 3) Both pairs of opposite sides are parallel .
• 4) Opposite angles are congruent.
• 5) Diagonals bisect .

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Calcworkshop

How To Prove a Parallelogram? 17 Step-by-Step Examples For Mastery!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson, you’re going to learn the 6 ways to prove a parallelogram.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

More specifically, how do we prove a quadrilateral is a parallelogram?

Finally, you’ll learn how to complete the associated 2 column-proofs.

Let’s jump in!

6 Properties of Parallelograms Defined

Well, we must show one of the six basic properties of parallelograms to be true!

• Both pairs of opposite sides are parallel
• Both pairs of opposite sides are congruent
• Both pairs of opposite angles are congruent
• Diagonals bisect each other
• One angle is supplementary to both consecutive angles (same-side interior)
• One pair of opposite sides are congruent AND parallel

So we’re going to put on our thinking caps, and use our detective skills, as we set out to prove (show) that a quadrilateral is a parallelogram.

This means we are looking for whether or not both pairs of opposite sides of a quadrilateral are congruent. Because if they are then the figure is a parallelogram.

In addition, we may determine that both pairs of opposite sides are parallel, and once again, we have shown the quadrilateral to be a parallelogram.

Opposite Sides Parallel and Congruent & Opposite Angles Congruent

Another approach might involve showing that the opposite angles of a quadrilateral are congruent or that the consecutive angles of a quadrilateral are supplementary. Both of these facts allow us to prove that the figure is indeed a parallelogram.

One Pair of Opposite Sides are Both Parallel and Congruent

Consecutive Angles in a Parallelogram are Supplementary

We might find that the information provided will indicate that the diagonals of the quadrilateral bisect each other. If so, then the figure is a parallelogram.

Diagonals of a Parallelogram Bisect Each Other

A tip from Math Bits says, if we can show that one set of opposite sides are both parallel and congruent, which in turn indicates that the polygon is a parallelogram, this will save time when working a proof.

In the video below:

• We will use the properties of parallelograms to determine if we have enough information to prove a given quadrilateral is a parallelogram.
• Find missing values of a given parallelogram.
• Write several two-column proofs (step-by-step).

Proving Parallelograms – Lesson & Examples (Video)

• Introduction to Proving Parallelograms
• 00:00:24 – How to prove a quadrilateral is a parallelogram? (Examples #1-6)
• Exclusive Content for Member’s Only
• 00:09:14 – Decide if you are given enough information to prove that the quadrilateral is a parallelogram. (Examples #7-13)
• 00:15:24 – Find the value of x in the parallelogram. (Examples #14-15)
• 00:18:36 – Complete the two-column proof. (Examples #16-17)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Table of Contents

Last modified on August 3rd, 2023

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6 ways of proving a quadrilateral is a parallelogram.

A quadrilateral is any two-dimensional flat shape having four sides. A parallelogram, on the other hand, is a quadrilateral having two pairs of opposite parallel sides.

To prove whether a given quadrilateral is a parallelogram, there are six possible ways. Depending upon the information provided, you need to use any one of the below-given properties of a parallelogram to get to your conclusion.

• Proving that both pairs of opposite sides are parallel
• Proving that both pairs of opposite sides are congruent
• Proving that one pair of opposite sides is both congruent and parallel
• Proving that the diagonals bisect each other
• Proving that one angle is supplementary to both consecutive angles
• Proving that both the pairs of opposite angles are congruent

If we can prove one of the above properties to be true about the given quadrilateral, we can conclude that the given figure is a parallelogram.  Also, it proves that all the six given properties are true for the given parallelogram.

Let us proof how a quadrilateral is a parallelogram.

Below given is a quadrilateral PQRS, whose opposite sides are parallel and congruent. We need to prove if the given quadrilateral is indeed a parallelogram. From the given information we understand that we need to prove the given quadrilateral is a parallelogram using property 1 and property 2.

Let us prove that the quadrilateral PQRS is a parallelogram. We will be using a two-column proof form (step by step) to get to our conclusion.

To Prove : The given quadrilateral PQRS, having one pair of opposite sides parallel and congruent, is a parallelogram.

Given : PS ≅ QR, PS ∥ QR

Let us draw two diagonal lines PR and QS

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5.19: Quadrilateral Classification in the Coordinate Plane

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Identify and calculate area of shapes based on coordinates on a plane.

Quadrilateral Classification

What if you were given a quadrilateral in the coordinate plane? How could you determine if that quadrilateral qualifies as one of the special quadrilaterals: parallelograms, squares, rectangles, rhombuses, kites, or trapezoids?

When working in the coordinate plane, you will sometimes want to know what type of shape a given shape is. You should easily be able to tell that it is a quadrilateral if it has four sides. But how can you classify it beyond that?

First you should graph the shape if it has not already been graphed. Look at it and see if it looks like any special quadrilateral. Do the sides appear to be congruent? Do they meet at right angles? This will give you a place to start.

Once you have a guess for what type of quadrilateral it is, your job is to prove your guess. To prove that a quadrilateral is a parallelogram , rectangle , rhombus , square, kite or trapezoid , you must show that it meets the definition of that shape OR that it has properties that only that shape has.

If it turns out that your guess was wrong because the shape does not fulfill the necessary properties, you can guess again. If it appears to be no type of special quadrilateral then it is simply a quadrilateral .

The examples below will help you to see what this process might look like.

Classifying Parallelograms

Determine what type of parallelogram TUNE is: \(T(0,10),\: U(4,2),\: N(−2,−1),\: and\: E(−6,7)\).

This looks like a rectangle. Let’s see if the diagonals are equal. If they are, then TUNE is a rectangle.

\(\begin{aligned} E U &=\sqrt{(-6-4)^{2}+(7-2)^{2}} & T N &=\sqrt{(0+2)^{2}+(10+1)^{2}} \\ &=\sqrt{(-10)^{2}+5^{2}} & &=\sqrt{2^{2}+11^{2}} \\ &=\sqrt{100+25} & &=\sqrt{4+121} \\ &=\sqrt{125} & &=\sqrt{125} \end{aligned}\)

If the diagonals are also perpendicular, then \(TUNE\) is a square.

\(\text { Slope of } E U=\dfrac{7-2}{-6-4}=-\dfrac{5}{10}=-\dfrac{1}{2} \quad \text { Slope of } T N=\dfrac{10-(-1)}{0-(-2)}=\dfrac{11}{2}\)

The slope of \(EU\) \(\neq\) slope of \(TN\), so \(TUNE\) is a rectangle.

Determining if a Quadrilateral is a Parallelogram

A quadrilateral is defined by the four lines \(y=2x+1\), \(y=−x+5\), \(y=2x−4\), and \(y=−x−5\). Is this quadrilateral a parallelogram?

To check if its a parallelogram we have to check that it has two pairs of parallel sides. From the equations we can see that the slopes of the lines are 2, −1, 2 and −1. Because two pairs of slopes match, this shape has two pairs of parallel sides and is a parallelogram.

Determining Types of Quadrilaterals

Determine what type of quadrilateral \(RSTV\) is. Simplify all radicals.

There are two directions you could take here. First, you could determine if the diagonals bisect each other. If they do, then it is a parallelogram. Or, you could find the lengths of all the sides. Let’s do this option.

\(\begin{aligned} R S &=\sqrt{(-5-2)^{2}+(7-6)^{2}} & S T &=\sqrt{(2-5)^{2}+(6-(-3))^{2}} \\ &=\sqrt{(-7)^{2}+1^{2}} & &=\sqrt{(-3)^{2}+9^{2}} \\ &=\sqrt{50}=5 \sqrt{2} & &=\sqrt{90}=3 \sqrt{10} \\ R V=& \sqrt{(-5-(-4))^{2}+(7-0)^{2}} & V T &=\sqrt{(-4-5)^{2}+(0-(-3))^{2}} \\ =& \sqrt{(-1)^{2}+7^{2}} & &=\sqrt{(-9)^{2}+3^{2}} \\ =& \sqrt{50}=5 \sqrt{2} & &=\sqrt{90}=3 \sqrt{10} \end{aligned}\)

From this we see that the adjacent sides are congruent. Therefore, \(RSTV\)is a kite.

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Algebra Review: When asked to “simplify the radical,” pull all square numbers (1, 4, 9, 16, 25, ...) out of the radical. Above \(\sqrt{50}=\sqrt{25\cdot 2}\). We know \(\sqrt{25}=5\), so \(\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}\).

Recognizing Parallelograms

Is the quadrilateral \(ABCD\) a parallelogram?

We have determined there are four different ways to show a quadrilateral is a parallelogram in the \(x−y\) plane. Let's check if a pair of opposite sides are congruent and parallel. First, find the length of \(AB\) and \(CD\).

\(\begin{aligned} A B &=\sqrt{(-1-3)^{2}+(5-3)^{2}} & C D &=\sqrt{(2-6)^{2}+(-2+4)^{2}} \\ &=\sqrt{(-4)^{2}+2^{2}} & &=\sqrt{(-4)^{2}+2^{2}} \\ &=\sqrt{16+4} & &=\sqrt{16+4} \\ &=\sqrt{20} & &=\sqrt{20} \end{aligned}\)

\(AB=CD\), so if the two lines have the same slope, \(ABCD\) is a parallelogram.

\(\text { Slope } A B=\dfrac{5-3}{-1-3}=\dfrac{2}{-4}=-\dfrac{1}{2} \text { Slope } C D=\dfrac{-2+4}{2-6}=\dfrac{2}{-4}=-\dfrac{1}{2}\)

Therefore, \(ABCD\) is a parallelogram.

Example \(\PageIndex{1}\)

A quadrilateral is defined by the four lines \(y=2x+1\), \(y=−2x+5\), \(y=2x−4\), and \(y=−2x−5\). Is this quadrilateral a rectangle?

To be a rectangle a shape must have four right angles. This means that the sides must be perpendicular to each other. From the given equations we see that the slopes are 2,−2, 2, and −2. Because the slopes are not opposite reciprocals of each other, the sides are not perpendicular, and the shape is not a rectangle.

Example \(\PageIndex{2}\)

Determine what type of quadrilateral \(ABCD\) is. \(A(−3,3),\: B(1,5),\: C(4,−1),\: D(1,−5)\). Simplify all radicals.

First, graph \(ABCD\). This will make it easier to figure out what type of quadrilateral it is. From the graph, we can tell this is not a parallelogram. Find the slope of \overline{BC} and \overline{AD} to see if they are parallel.

\(\begin{array}{l} \text { Slope of } \overline{B C}=\dfrac{5-(-1)}{1-4}=\dfrac{6}{-3}=-2 \\ \text { Slope of } \overline{A D}=\dfrac{3-(-5)}{-3-1}=\dfrac{8}{-4}=-2 \end{array}\)

We now know \(\overline{BC} \parallel \overline{AD}\). This is a trapezoid. To determine if it is an isosceles trapezoid, find \(AB\) and \(CD\).

\(\begin{aligned} A B &=\sqrt{(-3-1)^{2}+(3-5)^{2}} & S T &=\sqrt{(4-1)^{2}+(-1-(-5))^{2}} \\ &=\sqrt{(-4)^{2}+(-2)^{2}} & &=\sqrt{3^{2}+4^{2}} \\ &=\sqrt{20}=2 \sqrt{5} & &=\sqrt{25}=5 \end{aligned}\)

\(AB\neq CD\), therefore this is only a trapezoid.

Example \(\PageIndex{3}\)

Determine what type of quadrilateral \(EFGH\) is. \(E(5,−1),\: F(11,−3),\: G(5,−5),\: H(−1,−3)\)

We will not graph this example. Let’s find the length of all four sides.

\(\begin{aligned} E F &=\sqrt{(5-11)^{2}+(-1-(-3))^{2}} & F G &=\sqrt{(11-5)^{2}+(-3-(-5))^{2}} \\ &=\sqrt{(-6)^{2}+2^{2}} &=& \sqrt{6^{2}+2^{2}} \\ &=\sqrt{40}=2 \sqrt{10} & &=\sqrt{40}=2 \sqrt{10} \\ G H &=\sqrt{(5-(-1))^{2}+(-5-(-3))^{2}} & H &=\sqrt{(-1-5)^{2}+(-3-(-1))^{2}} \\ &=\sqrt{6^{2}+(-2)^{2}} & &=\sqrt{(-6)^{2}+(-2)^{2}} \\ =& \sqrt{40}=2 \sqrt{10} & &=\sqrt{40}=2 \sqrt{10} \end{aligned}\)

All four sides are equal. That means, this quadrilateral is either a rhombus or a square. The difference between the two is that a square has four 90∘ angles and congruent diagonals. Let’s find the length of the diagonals.

\(\begin{aligned} E G &=\sqrt{(5-5)^{2}+(-1-(-5))^{2}} & F H &=\sqrt{(11-(-1))^{2}+(-3-(-3))^{2}} \\ &=\sqrt{0^{2}+4^{2}} & &=\sqrt{12^{2}+0^{2}} \\ &=\sqrt{16}=4 & &=\sqrt{144}=12 \end{aligned}\)

The diagonals are not congruent, so \(EFGH\) is a rhombus.

• If a quadrilateral has exactly one pair of parallel sides, what type of quadrilateral is it?
• If a quadrilateral has two pairs of parallel sides and one right angle, what type of quadrilateral is it?
• If a quadrilateral has perpendicular diagonals, what type of quadrilateral is it?
• If a quadrilateral has diagonals that are perpendicular and congruent, what type of quadrilateral is it?
• If a quadrilateral has four congruent sides and one right angle, what type of quadrilateral is it?

Determine what type of quadrilateral \(ABCD\) is.

• \(A(−2,4),\: B(−1,2),\: C(−3,1),\: D(−4,3)\)
• \(A(−2,3),\: B(3,4),\: C(2,−1),\: D(−3,−2)\)
• \(A(1,−1),\: B(7,1),\: C(8,−2),\: D(2,−4)\)
• \(A(10,4),\: B(8,−2),\: C(2,2),\: D(4,8)\)
• \(A(0,0),\: B(5,0),\: C(0,4),\: D(5,4)\)
• \(A(−1,0),\: B(0,1),\: C(1,0),\: D(0,−1)\)
• \(A(2,0),\: B(3,5),\: C(5,0),\: D(6,5)\)

SRUE is a rectangle and PRUC is a square.

• What type of quadrilateral is \(SPCE\)?
• If \(SR=20\) and \(RU=12\), find \(CE\).
• Find SC and RC based on the information from part b. Round your answers to the nearest hundredth.

Review (Answers)

To view the Review answers, open this PDF file and look for section 6.8.

Proving that a Quadrilateral is a Parallelogram

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These Proving that a Quadrilateral is a Parallelogram lesson notes and worksheets cover:

• 6 properties that prove a quadrilateral is a parallelogram

This resource includes:

• 3 pages of notes (including 11 example problems and 2 two-column proofs) 1 problem involves showing a quadrilateral is a parallelogram on a coordinate plane
• 3 Practice worksheets The first and second include 8 problems each The third includes 1 coordinate plane problem and 2 two-column proofs

*Look at the preview to see exactly what is included!*

*I have included a 2nd version that leaves the proofs completely blank for students to fill in, depending on your needs!*

The practice worksheets would be perfect for in class practice or for homework!

Answer keys are included.

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