5.1 Quadrilaterals. Quadrilateral:_________________________ Parallelogram:_______________________.
6.3 Proving Quadrilaterals are Parallelograms · how to prove that a quadrilateral is a ... Ex. 4:...
Transcript of 6.3 Proving Quadrilaterals are Parallelograms · how to prove that a quadrilateral is a ... Ex. 4:...
Bell Ringer –
PQRS is a parallelogram.
If mP = 83 Degrees, then
Find the angle measure.
a. mR
b. mQ
P
R Q
83° S
Objectives:
Daily L:earning Target (DLT)
Friday November 30, 2012 “I can understand, apply, and remember
how to prove that a quadrilateral is a
parallelogram.”
Bell Ringer –
PQRS is a parallelogram.
If mP = 83 Degrees, then
Find the angle measure.
a. mR = 83
b. mQ = 97
P
R Q
83° S
Assignment 1. x = 5 9. It’s a
parallelogram
2. x = 3, y = 4 18. D
3. x = 1.6, y = 1 29. C
4. x = 5/3 30. F
5. x = 5
6. x = 13
7. It’s a parallelogram
8. It’s NOT a parallelogram
Theorems
Theorem 6.6: If both
pairs of opposite
sides of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Theorems
Theorem 6.7: If both
pairs of opposite
angles of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Theorems
Theorem 6.8: If an
angle of a
quadrilateral is
supplementary
to both of its
consecutive
angles, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
x°
(180 – x)° x°
Theorems
Theorem 6.9: If the
diagonals of a
quadrilateral
bisect each other,
then the
quadrilateral is a
parallelogram. ABCD is a parallelogram.
A
D
B
C
Ex. 2: Proving Quadrilaterals are
Parallelograms
As the sewing box below is opened, the
trays are always parallel to each other.
Why?
2.75 in. 2.75 in.
2 in.
2 in.
Ex. 2: Proving Quadrilaterals are
Parallelograms
Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.
2.75 in. 2.75 in.
2 in.
2 in.
Another Theorem ~
Theorem 6.10—If one pair of opposite
sides of a quadrilateral are congruent and
parallel, then the quadrilateral is a
parallelogram.
ABCD is a
parallelogram.
A
B C
D
Objective 2: Using Coordinate Geometry
When a figure is in the coordinate plane,
you can use the Distance Formula (see—it
never goes away) to prove that sides are
congruent and you can use the slope
formula (see how you use this again?) to
prove sides are parallel.
Ex. 4: Using properties of parallelograms
Show that A(2, -1), B(1,
3), C(6, 5) and D(7,1)
are the vertices of a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Ex. 4: Using properties of parallelograms Method 1—Show that opposite
sides have the same slope, so they are parallel.
Slope of AB. 3-(-1) = - 4
1 - 2
Slope of CD. 1 – 5 = - 4
7 – 6
Slope of BC. 5 – 3 = 2
6 - 1 5
Slope of DA. - 1 – 1 = 2
2 - 7 5
AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Because opposite sides are
parallel, ABCD is a
parallelogram.
Ex. 4: Using properties of parallelograms
Method 2—Show that
opposite sides have the
same length.
AB=√(1 – 2)2 + [3 – (- 1)2] = √17
CD=√(7 – 6)2 + (1 - 5)2 = √17
BC=√(6 – 1)2 + (5 - 3)2 = √29
DA= √(2 – 7)2 + (-1 - 1)2 = √29
AB ≅ CD and BC ≅ DA.
Because both pairs of opposites
sides are congruent, ABCD is a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Ex. 4: Using properties of parallelograms
Method 3—Show that
one pair of opposite
sides is congruent and
parallel.
Slope of AB = Slope of CD
= -4
AB=CD = √17
AB and CD are congruent
and parallel, so ABCD is a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Quadrilateral Family Tree Construction Project -
Each Group Shall Create The Following
Quadrilaterals:
1. Quadrilateral any size you want
2. 5 x 2 Parallelogram
3. 6 x 2 Rectangle
4. 4 x 4 Square
5. 3 x 3 Rhombus
6. 3 x 4 Kite
7. Isosceles Trapezoid any size you want
8. 8 arrows needed for the Quad Tree
Quadrilateral Family Tree Construction Project
Rubric –
5 Points Each
1. Quadrilaterals Measure Correctly
2. Quadrilaterals Are Cut With Lines
Shown
3. Quadrilaterals Are Appropriately
Labeled
4. At Least 8 arrows are needed for your
Quadrilateral Tree Project
5. Projects Are Neat