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Parallelograms

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Parallelograms

What if you were told that FGHI is a parallelogram and you are given the length of FG and the measure of \angle F ? What can you determine about HI , \angle H , \angle G , and \angle I ? After completing this Concept, you'll be able to apply parallelogram theorems to answer such questions.

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CK-12 Parallelograms

Guidance

A parallelogram is a quadrilateral with two pairs of parallel sides.

Notice that each pair of sides is marked parallel (for the last two shapes, remember that when two lines are perpendicular to the same line then they are parallel). Parallelograms have a lot of interesting properties. (You can explore the properties of a parallelogram at: http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php )

Facts about Parallelograms

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

If then

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

If then

3) Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then all pairs of consecutive angles are supplementary.

If then

m \angle A + m \angle D = 180^\circ\\m \angle A + m \angle B = 180^\circ\\m \angle B + m \angle C = 180^\circ\\m \angle C + m \angle D = 180^\circ

4) Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.

If then

Example A

ABCD is a parallelogram. If m \angle A = 56^\circ , find the measure of the other angles.

First draw a picture. When labeling the vertices, the letters are listed, in order.

If m \angle A = 56^\circ , then m \angle C = 56^\circ by the Opposite Angles Theorem.

m \angle A + m \angle B & = 180^\circ \quad \text{by the Consecutive Angles Theorem.}\\56^\circ + m \angle B & = 180^\circ\\m \angle B & = 124^\circ \quad m \angle D = 124^\circ \quad \text{because it is an opposite angle to} \ \angle B.

Example B

Find the values of x and y .

Remember that opposite sides of a parallelogram are congruent. Set up equations and solve.

6x - 7 & = 2x + 9 &&	y + 3 = 12\\4x & = 16 && \qquad y = 9\\x & = 4

Example C

Prove the Opposite Sides Theorem.

Given : ABCD is a parallelogram with diagonal \overline{BD}

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

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

The proof of the Opposite Angles Theorem is almost identical. You will try this proof in the problem set.

CK-12 Parallelograms

Guided Practice

1. Show that the diagonals of FGHJ bisect each other.

2. SAND is a parallelogram, SY = 4x - 11 and YN = x + 10 . Solve for x .

3. Find the measures of a and b in the parallelogram below:

Answers:

1. Find the midpoint of each diagonal.

&\text{Midpoint of} \ \overline{FH}: \qquad \left ( \frac{-4 + 6 }{2}, \frac{5 - 4}{2} \right ) = (1, 0.5)\\&\text{Midpoint of} \ \overline{GJ}:  \qquad \left ( \frac{3 - 1}{2}, \frac{3 - 2}{2} \right ) = (1,0.5)

Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.

2. Because this is a parallelogram, the diagonals bisect each other and SY\cong YN .

SY & = YN\\4x - 11 & = x + 10\\3x & = 21\\x & = 7

3. Consecutive angles are supplementary so 127^\circ +m\angle b=180^\circ which means that m\angle b=53^\circ . a and b are alternate interior angles and since the lines are parallel (since its a parallelogram), that means that m\angle a =m\angle b=53^\circ .

Practice

ABCD is a parallelogram. Fill in the blanks below.

  1. If AB = 6 , then CD = ______.
  2. If AE = 4 , then AC = ______.
  3. If m \angle ADC = 80^\circ, m \angle DAB = ______.
  4. If m \angle BAC = 45^\circ, m \angle ACD = ______.
  5. If m \angle CBD = 62^\circ, m \angle ADB = ______.
  6. If DB = 16 , then DE = ______.
  7. If m \angle B = 72^\circ in parallelogram ABCD , find the other three angles.
  8. If m \angle S = 143^\circ in parallelogram PQRS , find the other three angles.
  9. If \overline{AB} \perp \overline{BC} in parallelogram ABCD , find the measure of all four angles.
  10. If m \angle F = x^\circ in parallelogram EFGH , find the other three angles.

For questions 11-18, find the values of the variable(s). All the figures below are parallelograms.

Use the parallelogram WAVE to find:

  1. m \angle AWE
  2. m \angle ESV
  3. m \angle WEA
  4. m \angle AVW

Find the point of intersection of the diagonals to see if EFGH is a parallelogram.

  1. E(-1, 3), F(3, 4), G(5, -1), H(1, -2)
  2. E(3, -2), F(7, 0), G(9, -4), H(5, -4)
  3. E(-6, 3), F(2, 5), G(6, -3), H(-4, -5)
  4. E(-2, -2), F(-4, -6), G(-6, -4), H(-4, 0)

Fill in the blanks in the proofs below.

  1. Opposite Angles Theorem

Given : ABCD is a parallelogram with diagonal \overline{BD}

Prove : \angle A \cong \angle C

Statement Reason
1. 1. Given
2. \overline{AB} \| \overline{DC},\overline{AD} \| \overline{BC} 2.
3. 3. Alternate Interior Angles Theorem
4. 4. Reflexive PoC
5. \triangle ABD \cong \triangle CDB 5.
6. \angle A \cong \angle C 6.
  1. Parallelogram Diagonals Theorem

Given : ABCD is a parallelogram with diagonals \overline{BD} and \overline{AC}

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

Statement Reason
1. 1.
2. 2. Definition of a parallelogram
3. 3. Alternate Interior Angles Theorem
4. \overline{AB} \cong \overline{DC} 4.
5. 5.
6. \overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB} 6.
  1. Find x, y^\circ, and z^\circ . (The two quadrilaterals with the same side are parallelograms.)

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