2. Aim: Properties of Square & Rhombus Course: Applied Geo. Do Now: Aim: What are the properties of a rhombus and a square? Find the length of AD in rectangle ABCD, if AB = 2 and diagonal BD = 4. A rectangle has 4 right angles. a 2 + b 2 = c 2 Pythagorean Theorem a 2 + b 2 = c 2 ABD is a right triangle 2 2 + x 2 = 4 2 4 + x 2 = 16 x 2 = 12

3. Aim: Properties of Square & Rhombus Course: Applied Geo. Properties of a Rhombus A rhombus has all the properties of a parallelogram, PLUS, A rhombus is equilateral. The diagonals of a rhombus are perpendicular to each other. A rhombus is a parallelogram that has two congruent consecutive sides. The diagonals of a rhombus bisect its angles

4. Aim: Properties of Square & Rhombus Course: Applied Geo. 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 

5. Aim: Properties of Square & Rhombus Course: Applied Geo. Model Problem Find the measures of the numbered angles in the rhombus 50 0 1 3 2 4  1 = 90 0 Diagonals of a rhombus are Perpendicular  2 = 50 0 Alternate Interior  s   3 = 50 0 Diagonals of rhombus bisect the  s  4 = 40 0 The sum of the  s of a  equal 180 0 The diagonals of a rhombus are perpendicular to each other. 90 0 50 0 40 0 90 0

6. Aim: Properties of Square & Rhombus Course: Applied Geo. Regents Question In the diagram below of rhombus ABCD, m ∠ C = 100. What is m ∠ DBC? 40 o

7. Aim: Properties of Square & Rhombus Course: Applied Geo. 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 2x = 90  x = 45 3z = 90  z = 30 x + y = 90  45 + y = 90  y = 45 Diagonals are 

8. Aim: Properties of Square & Rhombus Course: Applied Geo. Model Problem Explain how and why ABCD is a rhombus Given: ABCD is a parallelogram AB = 2x + 1, DC = 3x – 11, AD = x + 13 Plan: Show that 2 consecutive sides are congruent. (AB  AD) A B DC 1)Since ABCD is a parallelogram, opposite sides are equal in length. DC = AB 3x – 11 = 2x + 1 3x – 2x = 11 + 1 x = 12 2)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.

9. Aim: Properties of Square & Rhombus Course: Applied Geo. Do Now: Aim: What are the properties of a rhombus and a square? In rhombus KLMN, KL = 3x, LM = 2(x + 3). Find the length of each side of the rhombus. KN LM

10. Aim: Properties of Square & Rhombus Course: Applied Geo. Properties of a Square A square is a rectangle that has two congruent sides. C B D A A square has all the properties of a rectangle, PLUS, A square has all the properties of a rhombus.

11. Aim: Properties of Square & Rhombus Course: Applied Geo. 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 

12. Aim: Properties of Square & Rhombus Course: Applied Geo. 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

13. Aim: Properties of Square & Rhombus Course: Applied Geo. Model Problem Find the value of the variables. In a square: diagonals are  to each other diagonals bisect opposite angles In a square: diagonals are  to each other diagonals bisect opposite angles x = 5 1 9x9x6z6z  1 = 3y – 6 y = 32 x = 7.5

14. Aim: Properties of Square & Rhombus Course: Applied Geo. 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)a square is a rectangle b)a square is a rhombus c)a rhombus is a square d)a square is a parallelogram

15. Aim: Properties of Square & Rhombus Course: Applied Geo. 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.