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Quadrilaterals
Geometry Chapter 6
Name: ________________________
6.1 Classifying Quadrilaterals
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Kite
Trapezoid
Isosceles Trapezoid
Draw the quadrilateral hierarchy:
2. Find the values of both of the variables.
1. Determine the most precise name for the quadrilateral
with vertices Q 4,4, B 2,9, H 8,9, and A10,4 .
3. Find the values of the variables in the rhombus.
Then find the lengths of the sides.
page 1
6.7 Proofs Using Coordinate Geometry
Determine the most precise name for the quadrilateral with the following vertices: A3,5, B7,6, C6,2, D2,1
Trapezoid Midsegment Theorem
Find the value of the variable in each trapezoid.
1.
2.G
A
x
B
3.M
H
35
x-3
5
E
30
F
x
K
Q
L
10
5
D
5x
C
J
I
15
N
4
R
4
O
P
2x-4
2. Segment MN is the midsegment of the trapezoid,
-Find the coordinates of M and N
-Find midpoint of the midsegment
page 2
3.
4.
5.
page 3
6.6 Placing Figures in the Coordinate Plane
Convenient Location:
Draw the given shapes with generic coordinates. Use as few variables as possible.
Square
Kite
Isosceles Trapezoid
page 4
page 5
6.2 Properties of Parallelograms
If 3 (or more) parallel lines cut off congruent segments on one transversal, then they cut
off congruent segments on every transversal (draw):
Examples
1. Find the value of x in parallelogram ABCD. Then find mA.
2. Find the values of x and y in parallelogram KLMN.
page 6
3. In the figure, DH CG BF AE , AB = BC = CD = 2, and EF = 2.5. Find EH.
4. Find the value of y in
EFGH . Then find mE , mF , mG and mH .
5. Find the measures of the angles in the parallelogram
page 7
6.3 Proving that a Quadrilateral is a Parallelogram
A quadrilateral is a parallelogram if:
*
*
*
*
Examples
1. Decide whether the quadrilateral must be a parallelogram. Explain
2. Find values for x and y for which ABCD must be a parallelogram.
3. Given: mE  mG  75, mF  105
Prove: EFGH is a parallelogram.
page 8
4.
Find the values of a and c for which PQRS must be a parallelogram.
page 9
6.4 Special Parallelograms
Theorem 6-9
Theorem 6-10
Theorem 6-11
~The converses of each theorem are also true. (In a parallelogram, if…)
6.4 Examples
1. Find the measures of the numbered angles in the rhombus.
m1 
m2 
m3 
m4 
2. One diagonal of a rectangle has length 8x  2 . The other diagonal has length 5x  11 . Find the
length of each diagonal.
3.
Find the measures of the numbered angles in
the rhombus.
4. Find the length of the diagonals of rectangle
GFED if FD  5 y  9 and GE  y  5 .
m1 
m2 
m3 
m4 
page 10
5. The diagonals of ABCD are perpendicular. AB=16 cm and BC=8 cm. Can ABCD be a rhombus or rectangle?
Explain.
6. A parallelogram has angle measures of 30, 150, 30, 150. Can you conclude that it is a rhombus or a rectangle?
Explain.
7. Fill in the chart below with “yes” or “no” in each box.
Parallelogram Rhombus
All sides are 
Opposite sides are
All angles are right angles
Diagonals bisect each other
Each diagonal bisects opposite angles
Opposite sides are 
Diagonals are perpendicular
Consecutive angles are supplementary
Diagonals are 
Rectangle
Square
page 11
6.5 Trapezoids and Kites
Isosceles Trapezoid (draw):
*
*
*
Kite:
*
*
*
*
Examples
1. WXYZ is an isosceles trapezoid, and mX  156. Find mY , mZ , and mW .
2. Find m1, m2, and m3 in the kite.
3. In the isosceles trapezoid, mS  70. Find mP, mQ, and mR.
page 12
4. Find m1, m2, and m3 in the kite.
5. Find the value of x in the figures
page 13