1. Squares and Rhombi
Lesson 6-5

2. Vocabulary: Rhombus: A parallelogram with all four sides congruent.
Square:  A parallelogram with all four sides congruent and all four right angles.

3. A rhombus (plural is rhombi) is a parallelogram with all four sides congruent.  A rhombus has all the characteristics of a parallelogram, with two additional characteristics.

5. A) According to Theorem 6
A) According to Theorem 6.16, if the figure is a rhombus, then each diagonal bisects a pair of angles.  Since figure FGHJ is defined as a rhombus, this theorem applies.  So, if m∠FJH = 82, then m∠KJH + m∠KJF = 82.  Since GJ  bisects these two angles, m∠KJH = m∠KJF, so m∠KJH is half of 82, which is 41 degrees.
So m∠KJH + m∠KHJ + 90 = 180
41+ m∠KHJ = 90
So m∠KHJ = 49
B) The definition of a rhombus is a parallelogram with all four sides congruent, so GH ≅ JH, thus GH = JH, so x+9=5x-2.  Then we can solve it using algebra.
x+9=5x-2
-x -x
9= 4x-2 
+2   +2
11= 4x (divide both sides by 4)
So x= 11/4

6. A square  is a parallelogram with four congruent sides and four right angles.  It is also a rectangle and a rhombus because it satisfies all of those requirements as well.

7. All of the properties of parallelograms, rectangles, and rhombi also apply to squares.  
--the diagonals of a parallelogram bisect each other
--the diagonals of a rectangle are congruent.
--the diagonals of a rhombus are perpendicular

8. Since JKLM is a parallelogram,  we know that we can use the properties of parallelograms, rectangles, squares, and rhombi to determine its true name.  If ΔJKL is isosceles, then we know that side JK ≅ side KL.  According to Theorem 6.19, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. So, JKLM is a rhombus.

9. Step 1:  Draw a graph.  This will help you visualize the problem, help you make conjectures, and determine reasonableness of your answer.
Step 2: Ask yourself if you know what it looks like and how do you know. The figure looks like a parallelogram. Maybe a Rhombus, definitely not a square or rectangle.

10. Continue the previous example:
Step 3: Prove it is a parallelogram. I am going to check using the distance formula of all the sides. (Remember you can use slope, distance, or midpoint formulas).
Since all sides are congruent, it must be a Rhombus.
Step 4: Justify your answer. Since all four sides are congruent, this is not just a parallelogram, but also a Rhombus according to the definition of a rhombus.

11. Continue the previous example:
Step 5: Remember, just because it doesn’t LOOK like a square or a rectangle, doesn’t mean that we can assume that they are not.  We must PROVE it.
According to Theorem 6.13, if a parallelogram is a rectangle, then its diagonals are congruent.  And Theorem 6.20 states that a quadrilateral has to be BOTH a rhombus AND a rectangle to be a square.  So, using the Distance formula, we can determine if the diagonals are congruent.
Since 68 ≠256 , the diagonals are NOT congruent, so the figure CANNOT be a rectangle and therefore, cannot be a square.
So it is a Rhombus!!!