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Understanding Rhombuses, Rectangles, and Squares: Properties and Theorems

This lesson focuses on rhombuses, rectangles, and squares, exploring their unique properties and corollaries. Learn how to determine the values of variables within geometric equations, and understand critical theorems, such as the conditions for a parallelogram to be classified as a rhombus or a rectangle. Through examples and exercises, students will reinforce their understanding of quadrilaterals and apply their knowledge to solve problems involving these special parallelograms.

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Understanding Rhombuses, Rectangles, and Squares: Properties and Theorems

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  1. 6.4 Rhombuses, Rectangles, and Squares Day 4

  2. Review • Find the value of the variables. p h 52° (2p-14)° 50° 68° p + 50° + (2p – 14)° = 180° p + 2p + 50° - 14° = 180° 3p + 36° = 180° 3p = 144 ° p = 48 ° 52° + 68° + h = 180° 120° + h = 180 ° h = 60°

  3. Special Parallelograms • Rhombus • A rhombus is a parallelogram with four congruent sides.

  4. Special Parallelograms • Rectangle • A rectangle is a parallelogram with four right angles.

  5. Special Parallelogram • Square • A square is a parallelogram with four congruent sides and four right angles.

  6. Corollaries • Rhombus corollary • A quadrilateral is a rhombus if and only if it has four congruent sides. • Rectangle corollary • A quadrilateral is a rectangle if and only if it has four right angles. • Square corollary • A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  7. Example • PQRS is a rhombus. What is the value of b? Q 2b + 3 = 5b – 6 9 = 3b 3 = b P 2b + 3 R S 5b – 6

  8. Review • In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___ • 1 • 2 • 3 • 4 • 5 7f – 3 = 4f + 9 3f – 3 = 9 3f = 12 f = 4

  9. Example • PQRS is a rhombus. What is the value of b? Q 3b + 12 = 5b – 6 18 = 2b 9 = b P 3b + 12 R S 5b – 6

  10. Theorems for rhombus • A parallelogram is a rhombus if and only if its diagonals are perpendicular. • A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. L

  11. Theorem of rectangle • A parallelogram is a rectangle if and only if its diagonals are congruent. A B D C

  12. The diagonals are congruent Both pairs of opposite sides are congruent Both pairs of opposite sides are parallel All angles are congruent All sides are congruent Diagonals bisect the angles Parallelogram Rectangle Rhombus Square Match the properties of a quadrilateral B,D A,B,C,D A,B,C,D B,D C,D C

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