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Special Parallelograms: Properties and Identification, Study notes of Geometry

Definitions, theorems, and corollaries for rhombuses, rectangles, and squares, as well as ways to identify parallelograms. It includes exercises to determine if given quadrilaterals are parallelograms based on their properties.

Typology: Study notes

2021/2022

Uploaded on 09/27/2022

torley
torley 🇺🇸

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Special Parallelograms
Definitions
Name
Definition
Picture
Rhombus
Rectangle
Square
Theorems and Corollaries
Name
Theorem/Corollary
Picture
Rhombus Corollary
A quadrilateral is a rhombus if and only if
Rectangle Corollary
A quadrilateral is a rectangle if and only if
Square Corollary
A quadrilateral is a square if and only if
Perpendicular Diagonals
Theorem
A parallelogram is a rhombus if and only if
Bisecting Diagonals
Theorem
A parallelogram is a rhombus if and only if
Congruent Diagonals
Theorem
A parallelogram is a rectangle if and only if
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Special Parallelograms Definitions Name Definition Picture Rhombus Rectangle Square Theorems and Corollaries Name Theorem/Corollary Picture Rhombus Corollary A quadrilateral is a rhombus if and only if Rectangle Corollary A quadrilateral is a rectangle if and only if Square Corollary A quadrilateral is a square if and only if Perpendicular Diagonals Theorem A parallelogram is a rhombus if and only if Bisecting Diagonals Theorem A parallelogram is a rhombus if and only if Congruent Diagonals Theorem A parallelogram is a rectangle if and only if

Ways to Prove a Quadrilateral is a Parallelogram Use the information in the diagram above to answer the following questions.

  1. For what value of x is quadrilateral CDEF a parallelogram? 2. For what value of x is quadrilateral MNPQ a parallelogram? Determine whether each quadrilateral is a parallelogram (use the table above). Write YES if it is.
  2. 1 pair of opposite sides is parallel and it has 2 consecutive right angles.
  3. It has 4 right angles.
  4. 1 pair of opposite sides is parallel and congruent.
  5. 1 pair of opposite sides is parallel. The other pair of opposite sides is congruent.
  6. Consecutive angles are supplementary.
  7. The diagonals are perpendicular.

Review

Special Parallelograms Parallelogram Rectangle Rhombus Square Trapezoid Sketch Properties 1 Both pair of opposite sides || 2 Both pair of consecutive sides ⊥ 3 Diagonals are ≅ 4 Diagonals are ⊥ 5 Diagonals bisect each other 6 Both diagonals are angle bisectors 7 All sides are ≅ 8 Exactly one pair of opposite sides are || 9 All angels are 90° 10 Both pair of opposite angles are ≅ 11 Consecutive angles are supplementary Examples: Name each quadrilateral – parallelogram, rectangle, rhombus, and square – for which the statement is true.

  1. It is equiangular.
  2. It is equilangular and equilateral.
  3. Its diagonals are perpendicular.
  4. Opposite angles are congruent.
  5. The diagonals bisect each other.
  6. The diagonals bisect opposite angles. Additional Examples: Name each quadrilateral and then find the values of x and y.