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Quadrilaterals that are Parallelograms

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Quadrilaterals that are Parallelograms

What if you were given four pairs of coordinates that form a quadrilateral? How could you determine if that quadrilateral is a parallelogram? After completing this Concept, you'll be able to use the Parallel Congruent Sides Theorem and other quadrilateral theorems to solve problems like this one.

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CK-12 Proving a Quadrilateral is a Parallelogram

Guidance

Recall that a parallelogram is a quadrilateral with two pairs of parallel sides. Even if a quadrilateral is not marked with having two pairs of sides, it still might be a parallelogram. The following is a list of theorems that will help you decide if a quadrilateral is a parallelogram or not.

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

If then

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

If then

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

If then

4) Parallel Congruent Sides Theorem: If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.

If then

You can use any of the above theorems to help show that a quadrilateral is a parallelogram. If you are working in the x-y plane, you might need to know the formulas shown below to help you use the theorems.

  • The Slope Formula, \frac{y_2 - y_1}{x_2 - x_1} . (Remember that if slopes are the same then lines are parallel).
  • The Distance Formula, \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} . (This will help you to show that two sides are congruent).
  • The Midpoint Formula, \left ( \frac{x_1 + x_2 }{2} , \frac{y_1 + y_2}{2} \right ) . (If the midpoints of the diagonals are the same then the diagonals bisect each other).

Example A

Prove the Opposite Sides Theorem Converse.

Given : \overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}

Prove : ABCD is a parallelogram

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

Example B

Is quadrilateral EFGH a parallelogram? How do you know?

a) By the Opposite Angles Theorem Converse, EFGH is a parallelogram.

b) EFGH is not a parallelogram because the diagonals do not bisect each other.

Example C

Is the quadrilateral ABCD a parallelogram?

Let’s use the Parallel Congruent Sides Theorem to see if ABCD is a parallelogram. First, find the length of AB and CD using the distance formula.

AB & = \sqrt{(-1-3)^2 + (5 - 3)^2} && CD = \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}

Next find the slopes to check if the lines are parallel.

\text{Slope}\ AB = \frac{5 - 3}{-1-3} = \frac{2}{-4} = -\frac{1}{2} \qquad \text{Slope}\ CD = \frac{-2 +4}{2-6} = \frac{2}{-4} = -\frac{1}{2}

AB = CD and the slopes are the same (implying that the lines are parallel), so ABCD is a parallelogram.

CK-12 Proving a Quadrilateral is a Parallelogram

Guided Practice

1. Prove the Parallel Congruent Sides Theorem.

Given : \overline{AB} \| \overline{DC} , and \overline{AB} \cong \overline{DC}

Prove : ABCD is a parallelogram

2. What value of x would make ABCD a parallelogram?

3. Is the quadrilateral RSTU a parallelogram?

Answers:

1.

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

2. \overline{AB} \| \overline{DC} . By the Parallel Congruent Sides Theorem, ABCD would be a parallelogram if AB = DC .

5x - 8 & = 2x + 13\\3x & = 21\\x & = 7

3. Let’s use the Parallelogram Diagonals Converse to see if RSTU is a parallelogram. Find the midpoint of each diagonal.

&\text{Midpoint of}\ RT = \left ( \frac{-4 + 3}{2},\frac{3 - 4}{2}\right ) = (-0.5,-0.5)\\&\text{Midpoint of}\ SU = \left ( \frac{4 - 5}{2}, \frac{5 - 5}{2} \right ) = (-0.5,0)

RSTU is not a parallelogram because the midpoints are not the same.

Practice

For questions 1-12, determine if the quadrilaterals are parallelograms.

For questions 13-18, determine the value of x and y that would make the quadrilateral a parallelogram.

For questions 19-22, determine if ABCD is a parallelogram.

  1. A(8, -1), B(6, 5), C(-7, 2), D(-5, -4)
  2. A(-5, 8), B(-2, 9), C(3, 4), D(0, 3)
  3. A(-2, 6), B(4, -4), C(13, -7), D(4, -10)
  4. A(-9, -1), B(-7, 5), C(3, 8), D(1, 2)

Fill in the blanks in the proofs below.

  1. Opposite Angles Theorem Converse

Given : \angle A \cong \angle C, \angle D \cong \angle B

Prove : ABCD is a parallelogram

Statement Reason
1. 1.
2. m \angle A = m \angle C, m \angle D = m \angle B 2.
3. 3. Definition of a quadrilateral
4. m \angle A + m \angle A + m \angle B + m \angle B = 360^\circ 4.
5. 5. Combine Like Terms
6. 6. Division PoE
7. \angle A and \angle B are supplementary \angle A and \angle D are supplementary 7.
8. 8. Consecutive Interior Angles Converse
9. ABCD is a parallelogram 9.
  1. Parallelogram Diagonals Theorem Converse

Given : \overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}

Prove : ABCD is a parallelogram

Statement Reason
1. 1.
2. 2. Vertical Angles Theorem
3. \triangle AED & \cong \triangle CEB\\\triangle AEB & \cong \triangle CED 3.
4. 4.
5. ABCD is a parallelogram 5.
  1. Given : \angle ADB \cong \angle CBD, \overline{AD} \cong \overline{BC} Prove : ABCD is a parallelogram

Statement Reason
1. 1.
2. \overline{AD} \| \overline{BC} 2.
3. ABCD is a parallelogram 3.

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