6 5 Conditions for Rhombuses Rectangles and Squares

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, a rectangle LDHE, it has 4 right s a rhombus GDEO,

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 diagonals, then the quadrilateral is a rhombus. R H O M B U S

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. R H O M B U S

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 congruent diagonals, then the quadrilateral is a rectangle. R E C T A N G L E

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. S Q U A R E

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 In Problem 1, how did you use the fact that ABCD is a parallelogram to prove that it is a rhombus? Explain. Theorem 6 -16: If a quadrilateral is a parallelogram with ⏊ diagonals, then the quadrilateral is a rhombus. � Another PROOF: Since ABCD is a , diagonals bisect each other. all 4 s 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? �

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. a counterexample Q: What statements are true about the diagonals of a 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. Using Properties of Special Parallelograms For what values of x and y is quadrilateral QRST a rhombus? Theorem: If a Solve for x: is a rhombus, then each diagonal is an bisector. Solve for y: 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. A: Only rhombuses or other parallelograms. 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; 2. In quadrilateral ABCD, m ABC = 56. What values of m 2, m 3, and m 4 ensure that quadrilateral 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. 3. A square has opposite vertices (-2, 1) and (2, 3). What are the other two vertices? Explain. A C

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. 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. It becomes a square if the diameters are ⏊.