Transcript 6-3-tests-for-parallelograms

Content Standards
G.CO.11 Prove theorems about
parallelograms.
G.GPE.4 Use coordinates to prove simple
geometric theorems algebraically.
Mathematical Practices
3 Construct viable arguments and critique
the reasoning of others.
2 Reason abstractly and quantitatively.
You recognized and applied properties of
parallelograms.
• Recognize the conditions that ensure a
quadrilateral is a parallelogram.
• Prove that a set of points forms a
parallelogram in the coordinate plane.
How we test if a quadrilateral is a parallelogram
1. Test if the opposite sides are congruent
2. Test if the opposite angles are
congruent
3. Test if diagonals bisect each other
(midpoint formula)
4. Test if the slopes of opposite sides are
the same (are opposite sides parallel?)
5. Determine if one pair of opposite sides
are congruent AND parallel
Identify Parallelograms
Determine whether the quadrilateral is a
parallelogram. Justify your answer.
Answer: Each pair of opposite sides has the same
measure. Therefore, they are congruent.
If both pairs of opposite sides of a
quadrilateral are congruent, the quadrilateral
is a parallelogram.
Which method would prove the
quadrilateral is a parallelogram?
A. Both pairs of opp. sides ||.
B. Both pairs of opp. sides .
C. Both pairs of opp. s .
D. One pair of opp. sides both
|| and .
Use Parallelograms to Prove
Relationships
MECHANICS Scissor lifts, like
the platform lift shown, are
commonly applied to tools
intended to lift heavy items. In the
diagram, A  C and B  D.
Explain why the consecutive
angles will always be
supplementary, regardless of the
height of the platform.
Use Parallelograms to Prove
Relationships
Answer: Since both pairs of opposite angles of
quadrilateral ABCD are congruent, ABCD is
a parallelogram by Theorem 6.10. Theorem
6.5 states that consecutive angles of
parallelograms are supplementary.
Therefore, mA + mB = 180 and
mC + mD = 180. By substitution,
mA + mD = 180 and mC + mB = 180.
The diagram shows a car jack used to raise a car
from the ground. In the diagram, AD  BC and
AB  DC. Based on this information, which
statement will be true, regardless of the height of
the car jack.
A. A  B
B. A  C
C. AB  BC
D. mA + mC = 180
Use Parallelograms and Algebra to Find Values
Find x and y so that the quadrilateral is a
parallelogram.
Use Parallelograms and Algebra to Find Values
AB = DC
Substitution
Distributive Property
Subtract 3x from each side.
Add 1 to each side.
Use Parallelograms and Algebra to Find Values
Substitution
Distributive Property
Subtract 3y from each side.
Add 2 to each side.
Answer: So, when x = 7 and y = 5, quadrilateral
ABCD is a parallelogram.
Find m so that the quadrilateral is a parallelogram.
A. m = 2
B. m = 3
C. m = 6
D. m = 8
Parallelograms and Coordinate Geometry
COORDINATE GEOMETRY
Quadrilateral QRST has vertices
Q(–1, 3), R(3, 1), S(2, –3), and
T(–2, –1). Determine whether the
quadrilateral is a parallelogram.
Justify your answer by using the
Slope Formula.
If the opposite sides of a quadrilateral are parallel,
then it is a parallelogram.
Parallelograms and Coordinate Geometry
Answer: Since opposite sides have the same slope,
QR║ST and RS║TQ. Therefore, QRST is a
parallelogram by definition.
Graph quadrilateral EFGH with vertices E(–2, 2),
F(2, 0), G(1, –5), and H(–3, –2). Determine whether
the quadrilateral is a parallelogram.
A. yes
B. no
Homework: Page 417 -418 #’s 1-7
Graph quadrilateral EFGH with vertices E(–2, 2),
F(2, 0), G(1, –5), and H(–3, –2). Determine whether
the quadrilateral is a parallelogram.
A. yes
B. no