a square  AFEO, a rectangle  LDHE, a rhombus  GDEO, 6-5 Conditions for Rhombuses, Rectangles, and Squares OBJECTIVES: To prove that a quadrilateral is a rhombus using sides, s, and diagonals; and apply properties of a rhombus to solve problems. Which vertices form a square? A rhombus? A rectangle? Justify your answers. a square  AFEO, it has 4 ≅ sides and 4 right s. a rectangle  LDHE, it has 4 right s a rhombus  GDEO, it has 4 ≅ sides of length 5.

RHOMBUS 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. RHOMBUS If a quadrilateral is a parallelogram with perpendicular diagonals, then the quadrilateral is a rhombus.

RHOMBUS 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. If a quadrilateral is a parallelogram with a diagonal that bisects a pair of opposite angles, then the quadrilateral is a rhombus. RHOMBUS

RECTANGLE 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. RECTANGLE If a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle.

SQUARE 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. If a quadrilateral is a parallelogram with perpendicular, congruent diagonals, then the quadrilateral is a square. SQUARE

⌍ 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. Proving Theorem 6-16 Theorem 6-16: If a quadrilateral is a parallelogram with ⏊ diagonals, then the quadrilateral is a rhombus. In Problem 1, how did you use the fact that ABCD is a parallelogram to prove that it is a rhombus? Explain. ⌍ PROOF: Since ABCD is a , 𝑨𝑪 and 𝑩𝑫 bisect each other, so 𝑩𝑬 ≅ 𝑫𝑬 . Since 𝑨𝑪 ⏊ 𝑩𝑫 , AED & AEB are ≅ right s. Another PROOF: Since ABCD is a , diagonals bisect each other. By the Reflexive Prop. of ≅, 𝑨𝑬 ≅ 𝑨𝑬 . 𝐒𝐨 𝑨𝑬 ≅ 𝑬𝑪 𝐚𝐧𝐝 𝑩𝑬 ≅ 𝑬𝑫 . Since 𝑨𝑪 ⏊ 𝑩𝑫 , all 4 s So AEB≅ AED by SAS. contain an included right , and are therefore all ≅. By CPCTC, 𝑨𝑩 ≅ 𝑫𝑬 . Since opposite sides of a are ≅, 𝑨𝑩 ≅ 𝑫𝑪 ≅ 𝑩𝑪 ≅ 𝑨𝑫 . By SAS, all 4 s are ≅. By CPCTC, 𝑨𝑩 ≅ 𝑫𝑪 ≅ 𝑩𝑪 ≅ 𝑨𝑫 . ∴ ABCD is a rhombus by definition . By definition, ABCD is a rhombus.

⌍ 6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. Proving Theorem 6-19 Why is showing that the diagonals of a quadrilateral are perpendicular bisectors not sufficient to prove the quadrilateral is a square? A: Rhombuses have diagonals that are ⏊ bisectors, but not all rhombuses are squares. ⌍ Q: Why is showing that the diagonals of a quadrilateral are ≅ not enough to prove the quadrilateral is a square? A: Rectangles have ≅ diagonals, but not all rectangles are squares.

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. Identifying Rhombuses, Rectangles, and Squares The diagonals of a quadrilateral are congruent. Can you conclude that the quadrilateral is a rectangle? Explain. No. A quadrilateral can have ≅ diagonals and not necessarily be a rectangle. The quadrilateral must be a with ≅ diagonals in order for it to be a rectangle. a counterexample Q: What statements are true about the diagonals of a rectangle? A: Rectangles have diagonals that bisect each other and are ≅.

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. Using Properties of Special Parallelograms For what values of x and y is quadrilateral QRST a rhombus? Theorem: If a is a rhombus, then each diagonal is an  bisector. Solve for x: Solve for y: 𝟒𝒙−𝟑𝟐=𝟐𝒙+𝟒 𝟐𝒚=𝒚+𝟐𝟓 𝒚=𝟐𝟓 2𝒙=𝟑𝟔 𝒙=𝟏𝟖 Take turns reading the question aloud. Discuss what the question is asking. Rephrase the question using a more familiar word order. Write out an equation to use as you solve the problem together.

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. Using Properties of Parallelograms In Problem 5, is there only one rectangle that can be formed by pulling the ropes taut? Explain. No, you can change the shape of the rectangle. Have two people holding different ropes move close together. Then have the other two people move until the ropes are taut again. Q: What kind of shape(s) can be formed if the ropes are made to bisect each other but are not ≅? A: Only rhombuses or other parallelograms. Q: What kind of shape(s) can be formed if the ropes are ≅ but do not bisect each other? A: Only quadrilaterals can be formed.

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. 1. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. Rhombus; Rectangle; diagonals are ⏊. diagonals are ≅. 2. In quadrilateral ABCD, mABC = 56. What values of m2, m3, and m4 ensure that quadrilateral ABCD is a rhombus? 𝟏 𝟐 𝟓𝟔 =𝟐𝟖 𝒎𝟏=𝒎𝟐=𝒎𝟑=𝒎𝟒=𝟐𝟖

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, s, & diagonals; and apply properties of a rhombus to solve problems. 3. A square has opposite vertices (-2, 1) and (2, 3). What are the other two vertices? Explain. Midpont of 𝑨𝑪  −𝟐+𝟐 𝟐 , 𝟏+𝟑 𝟐 𝒚  𝟎 , 𝟐 B −𝟏, 𝟒 C Slope of 𝑨𝑪  𝟐 𝟒 = 𝟏 𝟐 A C 𝟏, 𝟎 𝒙 Slope ⏊ to 𝑨𝑪  − 𝟐 𝟏 −𝟐 𝟐 Answer: −𝟏, 𝟒 and 𝟏 , 𝟎

6-5 Conditions for Rhombuses, Rectangles, and Squares To prove that a quadrilateral is a rhombus using sides, lengths, & diagonals; and apply properties of a rhombus to solve problems. 4. Explain Mathematical Ideas (1)(G) A quadrilateral has vertices 𝒙 𝟏 , 𝒚 𝟏 , 𝑬 𝒙 𝟐 , 𝒚 𝟐 , 𝑭 𝒙 𝟑 , 𝒚 𝟑 , 𝐚𝐧𝐝 𝑮 𝒙 𝟒 , 𝒚 𝟒 . Explain how you would determine algebraically whether the quadrilateral is a rhombus.  Show that opposite sides are ‖ by showing that their slopes are equal. Show that diagonals are ⏊ by showing that their slopes are negative reciprocals. 5. Analyze Mathematical Relationships (1)(F) Your friend says, “A parallelogram with perpendicular diagonals is a rectangle.” Is your friend correct? Explain. No; the only parallelogram with ⏊ diagonals are rhombuses and squares. 6. Connect Mathematical Ideas (1)(F) You draw a circle and two of its diameters. When you connect the endpoints of the diameters, what quadrilateral do you get? Explain. Rectangle; the diagonals are ≅ . 𝐚nd they bisect each other., It becomes a square if the diameters are ⏊.