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6.3: Proving Quadrilaterals are Parallelograms

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Learning Objectives

  • Prove a quadrilateral is a parallelogram.
  • Show a quadrilateral is a parallelogram in the x-y plane.

Review Queue

  1. Plot the points A(2, 2), B(4, -2), C(-2, -4), and D(-6, -2).
    1. Find the slopes of \overline{AB},\overline{BC}, \overline{CD}, and \overline{AD}. Is ABCD a parallelogram?
    2. Find the point of intersection of the diagonals by finding the midpoint of each.

Know What? You are marking out a baseball diamond and standing at home plate. 3^{rd} base is 90 feet away, 2^{nd} base is 127.3 feet away, and 1^{st} base is also 90 feet away. The angle at home plate is 90^\circ, from 1^{st} to 3^{rd} is 90^\circ. Find the length of the other diagonal (using the Pythagorean Theorem) and determine if the baseball diamond is a parallelogram.

Determining if a Quadrilateral is a Parallelogram

The converses of the theorems in the last section will now be used to see if a quadrilateral is a parallelogram.

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

If then

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

If then

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

If then

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

Example 1: Write a two-column proof.

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

Prove: ABCD is a parallelogram

Solution:

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

Theorem 6-10: If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.

If then

Example 2: Is quadrilateral EFGH a parallelogram? How do you know?

Solution:

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 3: Algebra Connection What value of x would make ABCD a parallelogram?

Solution: \overline{AB} \| \overline{DC}. By Theorem 6-10, ABCD would be a parallelogram if AB = DC.

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

Showing a Quadrilateral is a Parallelogram in the x-y Plane

To show that a quadrilateral is a parallelogram in the x-y plane, you might need:

  • The Slope Formula, \frac{y_2 - y_1}{x_2 - x_1}.
  • The Distance Formula, \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  • The Midpoint Formula, \left ( \frac{x_1 + x_2 }{2} , \frac{y_1 + y_2}{2} \right ).

Example 4: Is the quadrilateral ABCD a parallelogram?

Solution: Let’s use Theorem 6-10 to see if ABCD is a parallelogram. First, find the length of AB and CD.

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}

Find the slopes.

\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, ABCD is a parallelogram.

Example 5: Is the quadrilateral RSTU a parallelogram?

Solution: 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.

Know What? Revisited Use the Pythagorean Theorem to find the length of the second diagonal.

90^2 + 90^2 & = d^2\\8100 + 8100 & = d^2\\16200 & = d^2\\d & = 127.3

The diagonals are equal, so the other two sides of the diamond must also be 90 feet. The baseball diamond is a parallelogram, and more specifically, a square.

Review Questions

  • Questions 1-12 are similar to Example 2.
  • Questions 13-15 are similar to Example 3.
  • Questions 16-22 are similar to Examples 4 and 5.
  • Questions 23-25 are similar to Example 1 and the proof of the Opposite Sides Converse.

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

Algebra Connection 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.
2. m \angle A = m \angle C, m \angle D = m \angle B
3. Definition of a quadrilateral
4. m \angle A + m \angle A + m \angle B + m \angle B = 360^\circ
5. Combine Like Terms
6. Division PoE

7. \angle A and \angle B are supplementary

\angle A and \angle D are supplementary

8. Consecutive Interior Angles Converse
9. ABCD is a parallelogram
  1. Parallelogram Diagonals Theorem Converse

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

Prove: ABCD is a parallelogram

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

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

Review Queue Answers

1.

(a) \text{Slope}\ AB = \text{Slope} \ CD = -\frac{1}{2}\!\\{\;}\quad \ \text{Slope}\ AD = \text{Slope}\ BC = \frac{2}{3}\\

ABCD is a parallelogram because the opposite sides are parallel.

(b) \text{Midpoint of}\ BD = (0, -2)\!\\{\;}\quad \ \text{Midpoint of}\ AC = (0, -2)

Yes, the midpoints of the diagonals are the same, so they bisect each other.

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