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Lesson 8-7
Coordinate Proof with
Quadrilaterals
Transparency 8-7
5-Minute Check on Lesson 8-6
ABCD is an isosceles trapezoid with median EF.
1. Find mD if mA= 110°.
70°
2. Find x if AD = 3x² + 5 and BC = x² + 27.
±4
3. Find y if AC = 9(2y – 4) and BD = 10y + 12.
4. Find EF if AB = 10 and CD = 32.
B
A
E
F
D
C
6
21
5. Find AB if AB = r + 18, CD = 6r + 9 and EF = 4r + 10.
25
6. Standardized Test Practice: Which statement is always true about
trapezoid LMNO with bases of LM and NO?
A
LO // MN
B
LO  MN
C
LM // NO
D
LM  NO
Click the mouse button or press the
Space Bar to display the answers.
Objectives
• Position and label quadrilaterals for use in
coordinate proofs
• Prove theorems using coordinate proofs
Vocabulary
• Kite – quadrilateral with exactly two distinct
pairs of adjacent congruent sides.
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Rectangles
Rhombi
Squares
Kites
Trapezoids
Isosceles
Trapezoids
Name the missing coordinates for the isosceles trapezoid.
y
D(?, ?)
C(a-b, c)
x
A(0, 0)
B(a, 0)
The legs of an isosceles trapezoid are congruent and have
opposite slopes. Point C is c units up and b units to the left of B.
So, point D is c units up and b units to the right of A. Therefore,
the x-coordinate of D is 0 + c, or c and the y-coordinate of D is
0 + b, or b.
Answer: D (b, c)
Name the missing coordinates for the rhombus.
Answer:
Quadrilateral Characteristics Summary
Convex Quadrilaterals
Parallelograms
4 sided polygon
4 interior angles sum to 360
4 exterior angles sum to 360
Opposite sides parallel and congruent
Opposite angles congruent
Consecutive angles supplementary
Diagonals bisect each other
Rectangles
Trapezoids
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
Median is parallel to bases
Median = ½ (base + base)
Rhombi
Angles all 90°
Diagonals congruent
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Squares
Diagonals divide into 4 congruent triangles
Isosceles
Trapezoids
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
Do you know your characteristics?
• Extra Credit Assignment
• Review Problems
W
P
In the rectangle to the left,
WA = 6x, AH = 24, AHB = 33°,
WAP = y, and BAP = z – 5,
solve for x, y and z
A
B
H
R
In the square to the right,
RV = 5x, SV = 3y, VST = 9y,
and RS = z
solve for x, y and z
S
x=3
y=5
z = 15√2
V
U
J
T
K
N
L
x=4
y = 114°
z = 71°
M
In the rhombus to the left,
JK = 6x, KM = 2y, LNM = 10y,
JLN = 4z + 10, and JKN = 7z – 5,
solve for x, y and z
6x - 6
A
In the isosceles trapezoid to the right
EF is a median, solve for x, y and z
12z
E
y+4
21
6z
C 2x + 8 D
B
2y - 4
F
x=3
y=9
z=5
x=5
y=8
z = 10
Summary & Homework
• Summary:
– Position a quadrilateral so that a vertex is at the
origin and a least one side lies along an axis.
• Homework:
– pg 450-451; 9, 11-14, 28, 29, 31-33

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