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5.2 – Parallelograms

Key Terms

  • Bisect – To divide into two equal parts.
  • Consecutive Angles – Angles that are side-by-side.
  • Opposite Angles – Two angles that are not consecutive.
  • Opposite Sides – Two sides that are not consecutive.
  • Parallel – Lying in the same plane without intersecting.
    • The symbol || means “parallel.”
    • If parallel lines are graphed on a Cartesian coordinate system, they have the same slope.
  • Parallelogram – A quadrilateral in which the opposite sides are parallel and congruent

 

Review

  • Parallel Lines
    • They are coplanar (in the same plane).
    • They never intersect.
    • They are always the same distance apart.

GeoB 5.5 Parallel Lines

 

  • Quadrilateral – Polygon with 4 sides.

 

 


Notes

GeoB 5.2 Properties Parallelogram

 

  • In the parallelogram, ABCD (in the green box below), AB \cong DC and BC \cong AD
    • Also, AB || DC and BC || AD.

GeoB 5.5 Parallelograms

  •  In the above parallelogram, WXYZ, WX \cong ZY and XY \cong WZ

 


  • Proofs with Parallelograms (template below)

 

 

  • Template for Proofs

Geometric Proof Template


  • Example: Find the length of \overline{AD}

GeoB 5.5 Parallelogram Ex1

  • Remember: AD and BC are congruent, so set them equal to one another.
    • 14 + 2x = 6 + 6x
    • Subtract 2x on both sides: 14 = 6 + 4x
    • Subtract 6 on both sides: 8 = 4x
    • Divide by 4 on both sides: 2 = x
    • Substitute 2 into x for AD: 14 + 2(2)
    • Solve: 14 + 4 = 18

 


  • Example: Find x and y

GeoB 5.2 Consecutive Missing

  • Remember: Consecutive interior angles are supplementary (180°), so 110 + 2y must equal 180.
    • 110 + 2y = 180
    • Subtract 110 on both sides: 2y = 70
    • Divide by 2 on both sides: y = 35.
    • Since 2y = 70, we can replace 2y with 70 when we solve for x.
    • 70 = 5x
    • Divide by 5 on both sides: 14 = x.

 


  • Example: Given ABCD is a parallelogram, AC = 38, and AE = 3x + 4, find the value of x.

GeoB 5.2 Diagonals Example

  • Remember: Diagonals of a parallelogram BISECT each other; so, if AE = 3x + 4, then EC must also equal 3x + 4
  • Also, if AC is 38, then bisect it (cut it in half); so, AE = 19 and and EC = 19.
    • Set up: 3x + 4 = 19
    • Subtract 4 on both sides: 3x = 15
    • Divide by 3 on both sides: x = 5

 


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