6.4 Rhombuses, Rectangles and Squares Unit 1C3 Day 5.

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6.4 Rhombuses, Rectangles and Squares Unit 1C3 Day 5

Do Now:  A parallelogram ABCD has m  A = 3x + 15 and m  C = 5x – 17. What is the value of x?

Special Parallelograms

Relationships Between Parallelograms

Ex. 1: Describing a Special Parallelogram  Decide whether the statement is always, sometimes, or never true. a) A rhombus is a rectangle. b) A parallelogram is a rectangle. c) A rectangle is a parallelogram.

Ex. 2: Using properties of special parallelograms  ABCD is a rectangle. What else do you know about ABCD?

Sidebar: Biconditional Statements  If a statement and its converse are both true, then the statement is considered biconditional.  Instead of writing both the statement and its converse, we can combine the relationship into one statement using the phrase “if and only if.”  Ex. statement: If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.  Converse : If a quadrilateral has both pairs of opposite sides congruent, then it is a parallelogram.  Biconditional: A quadrilateral is a parallelogram if and only if both pairs of opposite sides are congruent.

Corollaries about special quadrilaterals:  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.

Ex. 3: Using properties of a Rhombus  In the diagram at the right, PQRS is a rhombus. What is the value of y?

Diagonals of special parallelograms  Theorem 6.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular.  ABCD is a rhombus if and only if AC | BD

Diagonals of special parallelograms  Theorem 6.12: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.  ABCD is a rhombus if and only if AC bisects  DAB and  BCD and BD bisects  ADC and  CBA.

Diagonals of special parallelograms  Theorem 6.13: A parallelogram is a rectangle if and only if its diagonals are congruent.  ABCD is a rectangle if and only if AC ≅ BD.

Ex 6: Checking a Rectangle (Carpentry) You are building a rectangular frame for a theater set. a. First, you nail four pieces of wood together as shown at the right. What is the shape of the frame? b. To make sure the frame is a rectangle, you measure the diagonals. One is 7 feet 4 inches. The other is 7 feet 2 inches. Is the frame a rectangle? Explain.

Closure  What is true of the diagonals of a rectangle and a square, but not of a rhombus?