6.4 Rhombuses, Rectangles and Squares

1. 6.4 Rhombuses, Rectangles and Squares Geometry NCSCOS: 1.02; 2.02; 2.03

2. U.E.Q What are the properties of different quadrilaterals? How do we use the formulas of areas of different quadrilaterals to solve real-life problems? Now a little review:

3. 6 6 Do you have enough information? Yes Yes No Yes

4. 50° 130° Do you have enough information? Yes No Yes Yes

5. Activator Test your prior knowledge and try to fill in the chart with properties of the following quadrilaterals:

6. Objectives: Use properties of sides and angles of rhombuses, rectangles, and squares. Use properties of diagonals of rhombuses, rectangles and squares.

7. E.Q What are properties of sides and angles of rhombuses, rectangles, and squares?

8. Properties of Special Parallelograms 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. In this lesson, you will study three special types of parallelograms: rhombuses, rectangles and squares.

9. Venn Diagram shows relationships-- MEMORIZE • Each shape has the properties of every group that it belongs to. For instance, a square is a rectangle, a rhombus and a parallelogram; so it has all of the properties of those shapes. parallelograms rhombuses rectangles squares

10. Some examples of a rhombus

11. Examples of rectangles

12. Examples of squares

13. Ex. 1: Describing a special parallelogram • Decide whether the statement is always, sometimes, or never true. • A rhombus is a rectangle. • A parallelogram is a rectangle. parallelograms rhombuses rectangles squares

14. Ex. 1: Describing a special parallelogram • Decide whether the statement is always, sometimes, or never true. • A rhombus is a rectangle. The statement is sometimes true. In the Venn diagram, the regions for rhombuses and rectangles overlap. IF the rhombus is a square, it is a rectangle. parallelograms rhombuses rectangles squares

15. Ex. 1: Describing a special parallelogram • Decide whether the statement is always, sometimes, or never true. • A parallelogram is a rectangle. The statement is sometimes true. Some parallelograms are rectangles. In the Venn diagram, you can see that some of the shapes in the parallelogram box are in the area for rectangles, but many aren’t. parallelograms rhombuses rectangles squares

16. Ex. 2: Using properties of special parallelograms • ABCD is a rectangle. What else do you know about ABCD? • Because ABCD is a rectangle, it has four right angles by definition. The definition also states that rectangles are parallelograms, so ABCD has all the properties of a parallelogram: • Opposite sides are parallel and congruent. • Opposite angles are congruent and consecutive angles are supplementary. • Diagonals bisect each other.

17. Take note: • A rectangle is defined as a parallelogram with four right angles. But any quadrilateral with four right angles is a rectangle because any quadrilateral with four right angles is a parallelogram. • 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. • You can use these to prove that a quadrilateral is a rhombus, rectangle or square without proving first that the quadrilateral is a parallelogram.

19. Ex. 3: Using properties of a Rhombus In the diagram at the right, PQRS is a rhombus. What is the value of y? All four sides of a rhombus are ≅, so RS = PS. 5y – 6 = 2y + 3 Equate lengths of ≅ sides. 5y = 2y + 9 Add 6 to each side. 3y = 9 Subtract 2y from each side. y = 3 Divide each side by 3.

20. Using diagonals of special parallelograms • The following theorems are about diagonals of rhombuses and rectangles. • 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.

21. Using 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.

22. Using diagonals of special parallelograms A B • 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. D C

23. NOTE: You can rewrite Theorem 6.11 as a conditional statement and its converse. Conditional statement: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Converse: If a parallelogram is a rhombus, then its diagonals are perpendicular.

24. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given

25. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given

26. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given • Def. of . Diagonals bisect each other.

27. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given • Def. of . Diagonals bisect each other. • Def. of . Diagonals bisect each other.

28. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given • Def. of . Diagonals bisect each other. • Def. of . Diagonals bisect each other. • SSS congruence post.

29. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given • Def. of . Diagonals bisect each other. • Def. of . Diagonals bisect each other. • SSS congruence post. • CPCTC

30. Ex. 4: Proving Theorem 6.11Given: ABCD is a rhombusProve: AC  BD Statements: • ABCD is a rhombus • AB ≅ CB • AX ≅ CX • BX ≅ DX • ∆AXB ≅ ∆CXB • AXB ≅ CXB • AC  BD Reasons: • Given • Given • Def. of . Diagonals bisect each other. • Def. of . Diagonals bisect each other. • SSS congruence post. • CPCTC • Congruent Adjacent s

31. Ex. 5: Coordinate Proof of Theorem 6.11Given: ABCD is a parallelogram, AC  BD.Prove: ABCD is a rhombus • Assign coordinates. Because AC BD, place ABCD in the coordinate plane so AC and BD lie on the axes and their intersection is at the origin. • Let (0, a) be the coordinates of A, and let (b, 0) be the coordinates of B. • Because ABCD is a parallelogram, the diagonals bisect each other and OA = OC. So, the coordinates of C are (0, - a). Similarly the coordinates of D are (- b, 0). A(0, a) D(- b, 0) B(b, 0) C(0, - a)

32. Ex. 5: Coordinate Proof of Theorem 6.11Given: ABCD is a parallelogram, AC  BD.Prove: ABCD is a rhombus • Find the lengths of the sides of ABCD. Use the distance formula (See – you’re never going to get rid of this) AB=√(b – 0)2 + (0 – a)2 = √b2 + a2 • BC= √(0 - b)2 + (– a - 0)2 = √b2 + a2 • CD= √(- b – 0)2 + [0 - (– a)]2 = √b2 + a2 • DA= √[(0 – (- b)]2 + (a – 0)2 = √b2 + a2 A(0, a) D(- b, 0) B(b, 0) C(0, - a) All the side lengths are equal, so ABCD is a rhombus.

33. Ex 6: Checking a rectangle 4 feet • CARPENTRY. You are building a rectangular frame for a theater set. • First, you nail four pieces of wood together as shown at the right. What is the shape of the frame? • 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. 6 feet 6 feet 4 feet

34. Ex 6: Checking a rectangle 4 feet • First, you nail four pieces of wood together as shown at the right. What is the shape of the frame? Opposite sides are congruent, so the frame is a parallelogram. 6 feet 6 feet 4 feet

35. Ex 6: Checking a rectangle 4 feet • 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. The parallelogram is NOT a rectangle. If it were a rectangle, the diagonals would be congruent. 6 feet 6 feet 4 feet

36. You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

37. Foldable 1. Take out a piece of notebook paper and make a hot dog fold over from the right side over to the pink line.

38. The fold crease Foldable 2. Now, divide the right hand section into 5 sections by drawing 4 evenly spaced lines. 3. Use scissors to cut along your drawn line, but ONLY to the crease!

39. The fold crease Foldable 4. Write QUADRILATERALS down the left hand side QUADRILATERALS

40. QUADRILATERALS The fold crease Foldable 5. Fold over the top cut section and write PARALLELOGRAM on the outside. Parallelogram 6. Reopen the fold.

41. QUADRILATERALS Foldable 7. On the left hand section, draw a parallelogram. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 8. On the right hand side, list all of the properties of a parallelogram.

42. QUADRILATERALS Foldable * Fold over the second cut section and write RECTANGLE on the outside. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel RECTANGLE * Reopen the fold.

43. QUADRILATERALS Foldable * On the left hand section, draw a rectangle. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 1. Special parallelogram. 2. Has 4 right angles 3. Diagonals are congruent. * On the right hand side, list all of the properties of a rectangle.

44. QUADRILATERALS Foldable * Fold over the third cut section and write RHOMBUS on the outside. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 1. Special parallelogram. 2. Has 4 right angles 3. Diagonals are congruent. * Reopen the fold. RHOMBUS

45. QUADRILATERALS Foldable * On the left hand section, draw a rhombus. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 1. Special parallelogram. 2. Has 4 right angles 3. Diagonals are congruent. * On the right hand side, list all of the properties of a rhombus. 1. Special Parallelogram 2. Has 4 Congruent sides 3. Diagonals are perpendicular. 4. Diagonals bisect opposite angles

46. QUADRILATERALS Foldable * Fold over the third cut section and write SQUARE on the outside. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 1. Special parallelogram. 2. Has 4 right angles 3. Diagonals are congruent. * Reopen the fold. 1. Special Parallelogram 2. Has 4 Congruent sides 3. Diagonals are perpendicular. 4. Diagonals bisect opposite angles SQUARE

47. QUADRILATERALS Foldable * On the left hand section, draw a square. 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are congruent. 4. Diagonals bisect each other. 5. Opposite sides are parallel 1. Special parallelogram. 2. Has 4 right angles 3. Diagonals are congruent. * On the right hand side, list all of the properties of a square. 1. Special Parallelogram 2. Has 4 Congruent sides 3. Diagonals are perpendicular. 4. Diagonals bisect opposite angles * Place in your notebook and save for tomorrow. 1. All the properties of parallelogram, rectangle, and rhombus 2. 4 congruent sides and 4 right angles

48. Warm-Up Name the figure described. A quadrilateral that is both a rhombus and a rectangle. A quadrilateral with exactly one pair of parallel sides. A parallelogram with perpendicular diagonals

49. 6.5 Trapezoids and Kites Geometry NCSCOS:1.02; 2.02; 2.03

50. U.E.Q What are the properties of different quadrilaterals? How do we use the formulas of areas of different quadrilaterals to solve real-life problems?