Download 8.4 Rectangles, Rhombus, Squares

Geometry Chapter 8: Quadrilaterals 8.4-­‐ Properties of Rhombuses, Rectangles, and Squares SWBAT: use properties of rhombuses, rectangles, and squares. Common Core: G.CO.11, G.CO.13, G.SRT.5 “Do Now” If 𝑃𝑄𝑅𝑆 is a parallelogram, find the values of x and y. Rhombus Corollary: Theorem 8.11: A parallelogram is a rhombus if and only if A quadrilateral is a its diagonals are rhombus if and only if it ______________________________________. has four ___________________________ sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four ___________________________ angles. Theorem 8.12: A parallelogram is a rhombus if and only if each diagonal ______________________ a pair of opposite angles. Square Corollary: A quadrilateral is a square if Theorem 8.13: A parallelogram is a and only if it is a ___________________________ and a rectangle if and only ___________________________________________. if its diagonals are ______________________________. Geometry Chapter 8: Quadrilaterals Parallelograms
(opposite sides are parallel)
Example 1: Decide whether the statement is always, sometimes, or never true. 1. A square is a rectangle. 2. The diagonals of a rectangle bisect the opposite angles. 3. A rectangle is a rhombus. 4. The diagonals of a square are perpendicular. 5. A parallelogram is a rhombus. 6. A rectangle is a square. Example 2: Given: 𝐴𝐵𝐶𝐷 is a parallelogram. 𝐴𝐶 ⊥ 𝐵𝐷 Prove: 𝐴𝐵𝐶𝐷 is a rhombus Statements Reasons Geometry Chapter 8: Quadrilaterals Example 3: Given the quadrilateral, find the missing measurements. 1. Given rectangle 𝐴𝐵𝐶𝐷. Find the following. 2. Given rectangle 𝐽𝐾𝐿𝑀. Find the following. 3. Given rectangle 𝑊𝑋𝑌𝑍, find 𝑚∠𝑋𝑍𝑊. 4. Given rectangle 𝐺𝐻𝐼𝐽, find 𝑚∠𝐽𝐻𝐼. 5. If 𝑀𝑁𝑂𝑃 is a rhombus, find 𝑀𝑃. Geometry Chapter 8: Quadrilaterals 6. 𝐽𝐾𝐿𝑀 is a rhombus. Find the following. 7. The quadrilateral below is a rhombus. Find the following. 8. 𝑅𝑆𝑇𝑈 is a rhombus. Find the following. 9. If 𝐶𝐷𝐸𝐹 is a rhombus, find 𝑚∠𝐹𝐸𝐷. Geometry Chapter 8: Quadrilaterals 10. 11. If 𝑆𝑇𝑈𝑉 is a square with 𝑆𝑊 = 2𝑥 + 13 and 𝑊𝑈 = 8𝑥 − 41, find 𝑉𝑇. 12. If 𝐹𝐺𝐻𝐼 is a square, solve for x. 13. 𝑆𝑇𝑈𝑉 is a rectangle. If 𝑉𝑊 = 9𝑥 − 11 and 𝑆𝑈 = 16𝑥 − 12, find 𝑊𝑇. 14. 𝐶𝐷𝐸𝐹 is a rectangle. Find 𝐸𝐹. Geometry Chapter 8: Quadrilaterals Coordinate Geometry: Name that Quadrilateral 1. Calculate the slope of each side. a. If both pairs of opposite sides have the same _____________________ then the quadrilateral is a ______________________________________. b. If the slopes of the consecutive sides are negative ____________________________ then the consecutive sides are ____________________________________ making the parallelogram a _________________________________________. 2. Calculate the slopes of the diagonals. a. If the slopes of the diagonals are __________________________ reciprocals then the _________________________ are perpendicular making the parallelogram a ________________________________________. b. If the parallelogram is both a _________________________________ and a ______________________________ then the parallelogram is a __________________________________. Examples: Determine if the quadrilateral with given vertices is a parallelogram, rectangle, rhombus, or square. Show all work to support your answer and explain your conclusion. 1. 𝑀 −5, −1 , 𝐴 −1, 7 , 𝑇 7, 5 , 𝐻(5, −3) Parallelogram: __________ Rectangle: __________ Rhombus: __________ Square: __________ Explanation: Geometry Chapter 8: Quadrilaterals 2. 𝑀 −4,0 , 𝐴 1, 5 , 𝑇 5, 1 , 𝐻(0, −4) Parallelogram: __________ Rectangle: __________ Rhombus: __________ Square: __________ Explanation: 3. 𝑀 −3, 1 , 𝐴 1, 3 , 𝑇 5, 1 , 𝐻(1, −1) Parallelogram: __________ Rectangle: __________ Rhombus: __________ Square: __________ Explanation: Geometry Chapter 8: Quadrilaterals 4. 𝑀 −4, 1 , 𝐴 1, 4 , 𝑇 6, 1 , 𝐻(1, −2) Parallelogram: __________ Rectangle: __________ Rhombus: __________ Square: __________ Explanation: 5. 𝑀 −2, 4 , 𝐴 2, 5 , 𝑇 4, −3 , 𝐻(0, −4) Parallelogram: __________ Rectangle: __________ Rhombus: __________ Square: __________ Explanation: