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Special Parallelograms

This guide explores the properties of parallelograms, focusing on rectangles and rhombi. Learn how to determine if a parallelogram is a rectangle or rhombus by using the properties of their diagonals. Discover the characteristics of rectangles, including the congruence of diagonals, and the unique attributes of rhombi, such as perpendicular and congruent diagonals. Through examples, we find missing values and analyze given points to identify the types of parallelograms. Whether for study or reference, this resource encapsulates essential geometric principles.

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Special Parallelograms

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  1. Special Parallelograms

  2. SWBAT… • To use properties of diagonals of rhombi and rectangles • To determine whether a parallelogram is a rhombus or a rectangle.

  3. Rectangles… • A rectangle is a quadrilateral with four right angles • If both pairs of opposite angles are congruent, then it is a parallelogram

  4. Theorem • If a parallelogram is a rectangle, then its diagonals are congruent.

  5. Example 1 • Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x. S R T U

  6. Example 2 • Find x and y if MNPL is a rectangle. 6y + 2 5x + 8 3x + 2

  7. Theorem • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  8. Example 3 • Determine whether parallelogram ABCD is a rectangle, given A (-2, 1), B (4, 3), C (5, 0) and D (-1, -2).

  9. Properties of Rhombi • A rhombus is a quadrilateral with all 4 sides are congruent. • The diagonals of a rhombus are perpendicular and congruent.

  10. Theorems • If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus. • Each diagonal of a rhombus bisects a pair of opposite angles.

  11. Example 4 • Find y if angle 1 = y2 – 10. • Find <PNL if <MPL = 64 M L 1 Q N P

  12. Example 5 • Determine whether parallelogram ABCD is a rhombus, rectangle or a square for A(-4, -2), B(-2, 6), C(6, 4), D(4, -4). List all the apply.

  13. Find the values of V, W, X, Y and Z. SURE is a rectangle. • SR = 4v +2 EU = 6v – 8 • EC = 2w + 3 CU = 3w – 1 • m<1 = x m<2 = 2x • UR = 6y -7 SE = 4y – 1 • m<1 = 2z m<3 = 8z S U 2 c 3 1 R E

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