6.4C Rhombuses, Rectangles, and Squares
Objectives: G.CO.11: Prove theorems about parallelograms.
For the Board: You will be able to use the properties of sides and angles of rhombuses, rectangles, and
squares.
A square is a quadrilateral with four congruent sides and four right angles.
A square is a parallelogram because both pairs of opposite sides are congruent.
Therefore it has all the properties of a parallelogram.
A square is a rectangle because it has 4 right angles.
Therefore it has the additional properties of a rectangle.
A square is a rhombus because all 4 sides are congruent.
Therefore it has the additional properties of a rhombus.
Properties of a Square
1. Opposite sides are parallel.
2. Opposite sides are congruent
3. Opposite angles are congruent.
4. Adjacent angles are supplementary
5. The diagonals bisect each other.
Venn Diagram
6. All angles are right angles.
7. The diagonals are congruent.
8. All sides are congruent.
9. The diagonals are perpendicular
10. The diagonals bisect the opposite angles.
Parallelograms
Rectangles
Squares Rhombuses
Open the book to page 422 and read example 3.
Example: Show that the diagonals of square EFGH are congruent,
perpendicular, and bisect each other.
FH =
0  12   4  32 
3  42  0  12 
F(-1, 3)
1  49  50
1  49  50
EG =
FH = EG
m(FH) = (-4 – 3)/(0 - -1) = -7
m(EG) = (0 - -1)/(3 –4) = 1/7
FH | EG
M(FH) = ((-1 + 0)/2, (3 + -4)/2) = (-1/2, -1/2)
M(EG) = ((-4 + 3)/2, (-1 + 0)/2) = (-1/2, -1/2)
FH and EG bisect each other.
G(3, 0)
E(-4, -1)
H(0, -4)
White Board Activity:
Practice: Show that the diagonals of square STVW are congruent,
perpendicular, and bisect each other.
S(-5, -4), T(0, 2), V(6, -3), W(1, -9)
SV =
TW =
6  52  (3  4) 2 
1  02   9  22 
T(0, 2)
121  1  122
1  121  122
SV = TW
m(SV) = (-3 + 4)/(6 + 5) = 1/11
m(TW) = (-9 -2)/(1 – 0) = -11/1 = -11
SV | TW
M(SV) = [(-5 + 6)/2, (-4 + -3)/2] = (½ , -7/2)
M(TW) =[(0 + 1)/2, (2 + -9)/2] = (½ , -7/2)
SV and TW bisect each other
S(-5, -4)
Practice: Decide whether the statement is always, sometimes, or never true.
a. A rectangle is a square.
Sometimes
b. A square is a rhombus.
Always
Assessment:
Student pairs will complete “CHECK IT OUT” prob. 3 from this section.
Independent Practice:
Text: pgs. 424 – 425 prob. 8, 16, 24 – 31.
For a Grade:
Text: pgs. 424 – 425 prob. 16, 24.
V(6, -3)
W(1, -9)