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Aim: What are the properties of a rhombus and a square?

Aim: What are the properties of a rhombus and a square?. Pythagorean Theorem. a 2 + b 2 = c 2. Do Now:. Find the length of AD in rectangle ABCD, if AB = 2 and diagonal BD = 4. ABD is a right triangle. A rectangle has 4 right angles. a 2 + b 2 = c 2. 2 2 + x 2 = 4 2.

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Aim: What are the properties of a rhombus and a square?

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  1. Aim: What are the properties of a rhombus and a square? Pythagorean Theorem a2 + b2 = c2 Do Now: Find the length of AD in rectangle ABCD, if AB = 2 and diagonal BD = 4. ABD is a right triangle A rectangle has 4 right angles. a2 + b2 = c2 22 + x2 = 42 4 + x2 = 16 x2 = 12

  2. Properties of a Rhombus A rhombus is a parallelogram that has two congruent consecutive sides. A rhombus has all the properties of a parallelogram, PLUS, A rhombus is equilateral. The diagonals of a rhombus are perpendicular to each other. The diagonals of a rhombus bisect its angles

  3. Properties of a Quadrilaterals Property Parallel. Rectangle Rhombus All sides are  Opposite sides are  Opposite sides are | | Opposite angles are  All angles are right  Diagonals bisect each other Diagonals are  Diagonals are  Each Diagonal bisects opposite 

  4. Model Problem 500 Find the measures of the numbered angles in the rhombus 900 1 900 500 3 2 500 4 400 The diagonals of a rhombus are perpendicular to each other. 1 = 900 Diagonals of a rhombus are Perpendicular 2 = 500 Alternate Interior s  3 = 500 Diagonals of rhombus bisect the s 4 = 400 The sum of the s of a D equal 1800

  5. Regents Question Inthe diagram below of rhombus ABCD, m∠C = 100. What is m∠DBC? 40o 40o

  6. Model Problem Find the value of the variables (2x)º (x + y)º (3z)º Since all sides are congruent this quadrilateral is a rhombus. Property Parallel. Rectangle Rhombus Diagonals are  2x = 90  x = 45 3z = 90  z = 30 x + y = 90  45 + y = 90  y = 45

  7. D C B A Model Problem Given: ABCD is a parallelogram AB = 2x + 1, DC = 3x – 11, AD = x + 13 Explain how and why ABCD is a rhombus Plan: Show that 2 consecutive sides are congruent. (AB  AD) Since ABCD is a parallelogram, opposite sides are equal in length. DC = AB 3x – 11 = 2x + 1 3x – 2x = 11 + 1 x = 12 Substitute x = 12 to find the length of AB and AD: AB = 2x + 1 = 2(12) + 1 = 25 AD = x + 13 = 12 + 13 = 25 AB  AD. Since parallelogram ABCD has two consecutive congruent sides, it’s a rhombus.

  8. Aim: What are the properties of a rhombus and a square? Do Now: In rhombus KLMN, KL = 3x, LM = 2(x + 3). Find the length of each side of the rhombus. L M K N

  9. Properties of a Square A B A square is a rectangle that has two congruent sides. D C A square has all the properties of a rectangle, PLUS, A square has all the properties of a rhombus.

  10. Properties of a Square Property Parallel. Rhombus Rectangle Square All sides are  Opposite sides are  Opposite sides are | | Opposite angles are  All angles are right  Diagonals bisect each other Diagonals are  Diagonals are  Each Diagonal bisects opposite 

  11. Model Problem ABCD is a square with diagonal BD. Determine if True or False. A. AB  BC B. AB  CD C. AB  AC D. 1  2 E. 1  3 F. B  4 G. ABC is isosceles H. ABC is right triangle I. ABC  ACD TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE

  12. Model Problem Find the value of the variables. 1 = 3y – 6 x = 5 y = 32 9x 6z 1 x = 7.5 In a square: diagonals are  to each other diagonals bisect opposite angles

  13. Model Problem ABCD is a square. If AB = 8x – 6 and BC = 5x + 12, find the length of each side of the square. • Which statement is false? • a square is a rectangle • a square is a rhombus • a rhombus is a square • a square is a parallelogram

  14. Model Problem In square ABCD diagonal AC is drawn. How many degrees are there in the measure of ACB? If the side of a square is 4, find the length of the diagonal.

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