6-4 Properties of Rhombuses, Rectangles, and Squares OBJECTIVES: To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides. A square is a parallelogram with four congruent sides and four right angles.

6-4 Properties of Rhombuses, Rectangles, and Squares

RECTANGLE 6-4 Properties of Rhombuses, Rectangles, and Squares OBJECTIVES: To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. RECTANGLE

6-4 Properties of Rhombuses, Rectangles, and Squares OBJECTIVES: To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. Fold a piece of notebook paper in half. Fold it in half again in the other direction. Draw a diagonal line from one vertex to the other. Cut through the folded paper along that line. Unfold the paper. What do you notice about the sides and about the diagonals of the figure you formed?  The sides are congruent.  The diagonals are ⏊ bisectors of each other.  Each diagonal bisects two angles.

6-4 Properties of Rhombuses, Rectangles, and Squares OBJECTIVES: To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. Use the figure in Problem 1. Is EFGH a rhombus, a rectangle, or a square? Explain.in  Opposite sides are ≅ in a , so all sides of EFGH are ≅.  EFGH a rhombus. Discuss with a classmate. Compare a rhombus, a rectangle, and a square. What do they have in common? How do they differ? Make a Venn diagram to show the relationships among these parallelograms.

6-4 Properties of Rhombuses, Rectangles, and Squares Use the list below to complete the concept map. parallelogram diagonals are congruent four right angles rhombus rhombus diagonals are perpendicular four right angles parallelogram square square diagonals are perpendicular diagonals are congruent

6-4 Properties of Rhombuses, Rectangles, and Squares OBJECTIVES: To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. Investigating Diagonals of Quadrilaterals a. Choose from a variety of tools (such as a protractor, a ruler, a compass, or a geoboard) to investigate patterns in the diagonals of squares. Explain your choice. Parallelogram Rectangle Rhombus easy to make different squares & diagonals Conjecture: The diagonals of a rectangle are congruent. Conjecture: The diagonals of a rhombus are perpendicular. Conjecture: The diagonals of a parallelogram bisect each other.

6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. b. Make several squares. Then make a conjecture about the diagonals of squares. Square Conjecture: The diagonals of a square are ⏊ and ≅.

6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. Finding Angle Measures What are the measures of the numbered angles in rhombus PQRS? 𝑚1=𝑚2= 𝑚3= 𝑚4= 38° Finding Diagonal Length a. If LN = 4x - 17 and MO = 2x + 13, what are the lengths of the diagonals of rectangle LMNO? LN = MO =43 b. What type of triangle is PMN? Explain. Diagonals of a rectangle are ≅and bisect each other,  PMN is isosceles.

6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. 1. What are the measures of the numbered angles in the rhombus? 𝑚1=40° 𝑚2=90° 𝑚3=50° 2. JKLM is a rectangle. If JL = 4x - 12 and MK = x, what is the value of x? What is the length of each diagonal? Solve for 𝒙: JL = MK MK = 4 𝟒𝒙−𝟏𝟐=𝒙 JL = 4 𝟑𝒙=𝟏𝟐 𝒙=𝟒

⌍ ⌌   6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. 3. A set designer prepares a plan for a backdrop in a play, as shown. According to the plan, ABCD is a rectangle and 𝐷𝐹 ≅ 𝐶𝐸 . Can the set designer conclude that ∆𝐴𝐷𝐸≅∆𝐵𝐶𝐹? Explain. YES, Since ABCD is a rectangle, 𝑨𝑫 ≅ 𝑩𝑪 𝐚𝐧𝐝 𝑫≅𝑪 because they are s.   ⌍ ⌌ By Segment Addition Postulate, 𝑫𝑬 ≅ 𝑪𝑭 . ∴∆𝑨𝑫𝑬≅∆𝑩𝑪𝑭 by SAS.

6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. 4. Vocabulary Which special parallelograms are equiangular? Which special parallelograms are equilateral? Equiangular  rectangle and square Equilateral  rhombus and square 5. Explain Mathematical Ideas (1)(G) Your class needs to find the value of x for which DEFG is a rectangle. A classmate’s work is shown below on the right. What is the error? Explain. ⇒ should have been ↳ (𝟐𝒙+𝟖)+(𝟗𝒙−𝟔)=𝟗𝟎

6-4 Properties of Rhombuses, Rectangles, and Squares To define and classify special types of parallelograms, as well as properties of diagonals of rhombuses and rectangles. 6. Analyze Mathematical Relationships (1)(F) Is it possible for a rhombus to be a rectangle? If so, what must be true about the rhombus? If not, why not? Yes; the rhombus must be a square.