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# 6.3: Proving Quadrilaterals are Parallelograms

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Prove a quadrilateral is a parallelogram using the converses of the theorems from the previous section.
• Prove a quadrilateral is a parallelogram in the coordinate plane.

## Review Queue

1. Write the converses of: the Opposite Sides Theorem, Opposite Angles Theorem, Consecutive Angles Theorem and the Parallelogram Diagonals Theorem.
2. Are any of these converses true? If not, find a counterexample.
3. Plot the points A(2,2), B(4,2), C(2,4)\begin{align*}A(2, 2), \ B(4, -2), \ C(-2, -4)\end{align*}, and D(6,2)\begin{align*}D(-6, -2)\end{align*}.
1. Find the slopes of AB¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯, CD¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}, \ \overline{BC}, \ \overline{CD}\end{align*}, and AD¯¯¯¯¯¯¯¯\begin{align*}\overline{AD}\end{align*}. Is ABCD\begin{align*}ABCD\end{align*} a parallelogram?
2. Find the point of intersection of the diagonals. Does this go along with what you found in part a?

Know What? Four friends, Geo, Trig, Algie, and Calc are marking out a baseball diamond. Geo is standing at home plate. Trig is 90 feet away at 3rd\begin{align*}3^{rd}\end{align*} base, Algie is 127.3 feet away at 2nd\begin{align*}2^{nd}\end{align*} base, and Calc is 90 feet away at 1st\begin{align*}1^{st}\end{align*} base. The angle at home plate is 90\begin{align*}90^\circ\end{align*}, from 1st\begin{align*}1^{st}\end{align*} to 3rd\begin{align*}3^{rd}\end{align*} is 90\begin{align*}90^\circ\end{align*}. Find the length of the other diagonal and determine if the baseball diamond is a parallelogram. If it is, what kind of parallelogram is it?

## Determining if a Quadrilateral is a Parallelogram

In the last section, we introduced the Opposite Sides Theorem, Opposite Angles Theorem, Consecutive Angles Theorem and the Parallelogram Diagonals Theorem. #1 in the Review Queue above, had you write the converses of each of these:

Opposite Sides Theorem Converse: If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.

Opposite Angles Theorem Converse: If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.

Consecutive Angles Theorem Converse: If the consecutive angles of a quadrilateral are supplementary, then the figure is a parallelogram.

Parallelogram Diagonals Theorem Converse: If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.

Are these converses true? The answer is yes. Each of these converses can be a way to show that a quadrilateral is a parallelogram. However, the Consecutive Angles Converse can be a bit tricky, considering you would have to show that each angle is supplementary to its neighbor (A\begin{align*}(\angle A\end{align*} and B, B\begin{align*}\angle B, \ \angle B\end{align*} and C, C\begin{align*}\angle C, \ \angle C\end{align*} and D\begin{align*}\angle D\end{align*}, and A\begin{align*}\angle A\end{align*} and D)\begin{align*}\angle D)\end{align*}. We will not use this converse.

Proof of the Opposite Sides Theorem Converse

Given: AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯, AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DC},\ \overline{AD} \cong \overline{BC}\end{align*}

Prove: ABCD\begin{align*}ABCD\end{align*} is a parallelogram

Statement Reason
1. AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯, AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DC}, \ \overline{AD} \cong \overline{BC}\end{align*} Given
2. DB¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯\begin{align*}\overline{DB} \cong \overline{DB}\end{align*} Reflexive PoC
3. ABDCDB\begin{align*}\triangle ABD \cong \triangle CDB\end{align*} SSS
4. ABDBDC, ADBDBC\begin{align*}\angle ABD \cong \angle BDC, \ \angle ADB \cong \angle DBC\end{align*} CPCTC
5. AB¯¯¯¯¯¯¯¯ || DC¯¯¯¯¯¯¯¯, AD¯¯¯¯¯¯¯¯ || BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \ || \ \overline{DC},\ \overline{AD} \ || \ \overline{BC}\end{align*} Alternate Interior Angles Converse
6. ABCD\begin{align*}ABCD\end{align*} is a parallelogram Definition of a parallelogram

Example 1: Write a two-column proof.

Given: AB¯¯¯¯¯¯¯¯ || DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \ || \ \overline{DC}\end{align*} and AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DC}\end{align*}

Prove: ABCD\begin{align*}ABCD\end{align*} is a parallelogram

Solution:

Statement Reason
1. AB¯¯¯¯¯¯¯¯ || DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \ || \ \overline{DC}\end{align*} and AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DC}\end{align*} Given
2. ABDBDC\begin{align*}\angle ABD \cong \angle BDC\end{align*} Alternate Interior Angles
3. DB¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯\begin{align*}\overline{DB} \cong \overline{DB}\end{align*} Reflexive PoC
4. ABDCDB\begin{align*}\triangle ABD \cong \triangle CDB\end{align*} SAS
5. AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AD} \cong \overline{BC}\end{align*} CPCTC
6. ABCD\begin{align*}ABCD\end{align*} is a parallelogram Opposite Sides Converse

Example 1 proves an additional way to show that a quadrilateral is a parallelogram.

Theorem 5-10: If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.

Example 2: Is quadrilateral EFGH\begin{align*}EFGH\end{align*} a parallelogram? How do you know?

Solution: For part a, the opposite angles are equal, so by the Opposite Angles Theorem Converse, EFGH\begin{align*}EFGH\end{align*} is a parallelogram. In part b, the diagonals do not bisect each other, so EFGH\begin{align*}EFGH\end{align*} is not a parallelogram.

Example 3: Algebra Connection What value of x\begin{align*}x\end{align*} would make ABCD\begin{align*}ABCD\end{align*} a parallelogram?

Solution: AB¯¯¯¯¯¯¯¯ || DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \ || \ \overline{DC}\end{align*} from the markings. By Theorem 5-10, ABCD\begin{align*}ABCD\end{align*} would be a parallelogram if AB=DC\begin{align*}AB =DC\end{align*} as well.

5x83xx=2x+13=21=7\begin{align*}5x-8&=2x+13\\ 3x&=21\\ x&=7\end{align*}

In order for ABCD\begin{align*}ABCD\end{align*} to be a parallelogram, x\begin{align*}x\end{align*} must equal 7.

## Showing a Quadrilateral is a Parallelogram in the Coordinate Plane

To show that a quadrilateral is a parallelogram in the xy\begin{align*}x-y\end{align*} plane, you will need to use a combination of the slope formulas, the distance formula and the midpoint formula. For example, to use the Definition of a Parallelogram, you would need to find the slope of all four sides to see if the opposite sides are parallel. To use the Opposite Sides Converse, you would have to find the length (using the distance formula) of each side to see if the opposite sides are congruent. To use the Parallelogram Diagonals Converse, you would need to use the midpoint formula for each diagonal to see if the midpoint is the same for both. Finally, you can use Theorem 5-10 in the coordinate plane. To use this theorem, you would need to show that one pair of opposite sides has the same slope (slope formula) and the same length (distance formula).

Example 4: Is the quadrilateral \begin{align*}ABCD\end{align*} a parallelogram?

Solution: We have determined there are four different ways to show a quadrilateral is a parallelogram in the \begin{align*}x-y\end{align*} plane. Let’s use Theorem 5-10. First, find the length of \begin{align*}AB\end{align*} and \begin{align*}CD\end{align*}.

\begin{align*}AB& =\sqrt{(-1-3)^2+(5-3)^2} && CD=\sqrt{(2-6)^2+(-2+4)^2}\\ & = \sqrt{(-4)^2+2^2} && \quad \ \ =\sqrt{(-4)^2+2^2}\\ & = \sqrt{16+4} && \quad \ \ =\sqrt{16+4}\\ & = \sqrt{20} && \quad \ \ =\sqrt{20}\end{align*}

\begin{align*}AB = CD\end{align*}, so if the two lines have the same slope, \begin{align*}ABCD\end{align*} is a parallelogram.

Slope \begin{align*}AB = \frac{5-3}{-1-3}=\frac{2}{-4}=-\frac{1}{2}\end{align*} Slope \begin{align*}CD = \frac{-2+4}{2-6}=\frac{2}{-4}=-\frac{1}{2}\end{align*}

By Theorem 5-10, \begin{align*}ABCD\end{align*} is a parallelogram.

Example 5: Is the quadrilateral \begin{align*}RSTU\end{align*} a parallelogram?

Solution: Let’s use the Parallelogram Diagonals Converse to determine if \begin{align*}RSTU\end{align*} is a parallelogram. Find the midpoint of each diagonal.

Midpoint of \begin{align*}RT = \left ( \frac{-4+3}{2}, \ \frac{3-4}{2} \right )=(-0.5,-0.5)\end{align*}

Midpoint of \begin{align*}SU = \left ( \frac{4-5}{2}, \ \frac{5-5}{2} \right ) =(-0.5,0)\end{align*}

Because the midpoint is not the same, \begin{align*}RSTU\end{align*} is not a parallelogram.

Know What? Revisited First, we can use the Pythagorean Theorem to find the length of the second diagonal.

\begin{align*}90^2+90^2&=d^2\\ 8100+8100&=d^2\\ 16200&=d^2\\ d&=127.3\end{align*}

This means that the diagonals are equal. If the diagonals are equal, the other two sides of the diamond are also 90 feet. Therefore, the baseball diamond is a parallelogram, and more specifically, it is a square.

## Review Questions

For questions 1-12, determine if the quadrilaterals are parallelograms. If they are, write a reason.

For questions 13-15, determine the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} that would make the quadrilateral a parallelogram.

For questions 16-18, determine if \begin{align*}ABCD\end{align*} is a parallelogram.

1. \begin{align*}A(8, -1), \ B(6, 5), \ C(-7, 2), \ D(-5, -4)\end{align*}
2. \begin{align*}A(-5, 8), \ B(-2, 9), \ C(3, 4), \ D(0, 3)\end{align*}
3. \begin{align*}A(-2, 6), \ B(4, -4), \ C(13, -7), \ D(4, -10)\end{align*}

Write a two-column proof.

1. Opposite Angles Theorem Converse Given: \begin{align*}\angle A \cong \angle C, \ \angle D \cong \angle B\end{align*} Prove: \begin{align*}ABCD\end{align*} is a parallelogram
2. Parallelogram Diagonals Theorem Converse Given: \begin{align*}\overline{AE} \cong \overline{EC}, \ \overline{DE} \cong \overline{EB}\end{align*} Prove: \begin{align*}ABCD\end{align*} is a parallelogram
3. \begin{align*}{\;}\end{align*} Given: \begin{align*}\angle ADB \cong CBD, \ \overline{AD} \cong \overline{BC}\end{align*} Prove: \begin{align*}ABCD\end{align*} is a parallelogram

Suppose that \begin{align*}A(-2, 3), \ B(3, 3)\end{align*} and \begin{align*}C(1, -3)\end{align*} are three of four vertices of a parallelogram.

1. Depending on where you choose to put point \begin{align*}D\end{align*}, the name of the parallelogram you draw will change. Sketch a picture to show all possible parallelograms. How many can you draw?
2. If you know the parallelogram is named \begin{align*}ABDC\end{align*}, what is the slope of side parallel to \begin{align*}\overline{AC}\end{align*}?
3. Again, assuming the parallelogram is named \begin{align*}ABDC\end{align*}, what is the length of \begin{align*}\overline{BD}\end{align*}?
4. Find the points of intersection of the diagonals of the three parallelograms formed. Label them \begin{align*}X\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, \begin{align*}Y\end{align*} in parallelogram \begin{align*}ADBC\end{align*} and \begin{align*}Z\end{align*} in parallelogram \begin{align*}ABDC\end{align*}.
5. Connect the points \begin{align*}X, \ Y\end{align*} and \begin{align*}Z\end{align*} to form a triangle. What do you notice about this triangle?

The points \begin{align*}Q(-1, 1), \ U(7, 1), \ A(1, 7)\end{align*} and \begin{align*}D(-1, 5)\end{align*} are the vertices of quadrilateral \begin{align*}QUAD\end{align*}. Plot the points on graph paper to complete problems 27-30.

1. Find the midpoints of sides \begin{align*}\overline{QU}, \ \overline{UA}, \ \overline{AD}\end{align*} and \begin{align*}\overline{DQ}\end{align*}. Label them \begin{align*}W, \ X, \ Y\end{align*} and \begin{align*}Z\end{align*} respectively.
2. Connect the midpoints to form quadrilateral \begin{align*}WXYZ\end{align*}. What does this quadrilateral appear to be?
5. This phenomenon occurs in all quadrilaterals. Describe how you might prove this fact. (Hint: each side of quadrilateral \begin{align*}WXYZ\end{align*} is a midsegment in a triangle formed by two sides of the parallelogram and a diagonal.)

1. Opposite Sides Theorem Converse: If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.

Opposite Angles Theorem Converse: If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.

Consecutive Angles Theorem Converse: If the consecutive angles of a quadrilateral are supplementary, then the figure is a parallelogram.

Parallelogram Diagonals Theorem Converse: If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.

2. All the converses are true.

3.

a) \begin{align*}\text{Slope}\ AB = \text{Slope} \ CD = -\frac{1}{2}\!\\ {\;} \ \ \text{Slope}\ AD = \text{Slope}\ BC = \frac{2}{3}\\\end{align*}

\begin{align*}ABCD\end{align*} is a parallelogram because the opposite sides are parallel.

b) \begin{align*}\text{Midpoint of}\ BD = (0, -2)\!\\ {\;} \ \ \text{Midpoint of}\ AC = (0, -2)\end{align*}

Yes, the midpoint of the diagonals are the same, so they bisect each other. This corresponds with what we found in part a.

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