Transcript 8.2 Parallelograms - St. Monica Catholic Church

STAND AND DELIVER QUIZ
8.2 PARALLELOGRAMS
Parallelogram- a quadrilateral with both pairs of opposite sides parallel.
Theorem 8.3- Opposite sides of a parallelogram are congruent
Theorem 8.4- Opposite angles in a parallelogram are congruent
Theorem 8.5- Consecutive angles in a parallelogram are supplementary
Theorem 8.6- If a parallelogram has one right angle, it has four right angles.
Theorem 8.7- The diagonals of a parallelogram bisect each other.
Theorem 8.8 Each diagonal of a parallelogram separates the parallelogram
into two congruent triangles.
PROPERTIES OF PARALLELOGRAMS
RSTU is a parallelogram. Find
and y.
If lines are cut by a transversal, alt. int.
Definition of congruent angles
Substitution
Angle Addition Theorem
Substitution
Subtract 58 from each
side.
Definition of congruent segments
Substitution
Divide each side by 3.
Answer:
ABCD is a parallelogram.
Answer:
DIAGONALS OF PARALLELOGRAMS
MULTIPLE-CHOICE TEST ITEM
What are the coordinates of the intersection of the diagonals of
parallelogram MNPR, with vertices
M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?
A
B
C
D
Solve the Test Item
Find the midpoint of
Midpoint Formula
The coordinates of the intersection of the diagonals of parallelogram
MNPR are (1, 2).
Answer: C
MULTIPLE-CHOICE TEST ITEM
What are the coordinates of the intersection of the diagonals of
parallelogram LMNO, with vertices
L(0, –3), M(–2, 1), N(1, 5), O(3, 1)?
A
Answer: B
B
C
D
8.3 TESTS FOR
PARALLELOGRAMS
Proving Parallelograms
Theorem 8.9- If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram
Theorem 8.10 If both pairs of opposite angles are congruent, then
the quadrilateral is a parallelogram
Theorem 8.11 If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram
Theorem 8.12 If one pair of opposite sides of a quadrilateral is both
parallel and congruent, then the quadrilateral is a parallelogram
WRITE A PROOF
Write a paragraph proof of the statement: If a diagonal of a
quadrilateral divides the quadrilateral into two congruent
triangles, then the quadrilateral is a parallelogram.
Given:
Prove: ABCD is a parallelogram.
Proof:
CPCTC. By Theorem 8.9, if both pairs of opposite sides of a
quadrilateral are congruent, the quadrilateral is a
parallelogram. Therefore, ABCD is a parallelogram.
Write a paragraph proof of the statement: If two diagonals of a
quadrilateral divide the quadrilateral into four triangles where
opposite triangles are congruent, then the quadrilateral is a
parallelogram.
Given:
Prove: WXYZ is a parallelogram.
Proof:
by CPCTC. By Theorem 8.9,
if both pairs of opposite sides of a quadrilateral are
congruent, the quadrilateral is a parallelogram. Therefore,
WXYZ is a parallelogram.
PROPERTIES OF
PARALLELOGRAMS
Some of the shapes in this
Bavarian crest appear to be
parallelograms. Describe the
information needed to determine
whether the shapes are
parallelograms.
Answer: If both pairs of opposite sides are the same length or if one pair
of opposite sides is a congruent and parallel, the
quadrilateral is a parallelogram. If both pairs of opposite
angles are congruent or if the diagonals bisect
each other, the quadrilateral is
a parallelogram.
The shapes in the vest pictured
here appear to be parallelograms.
Describe the information needed
to determine whether the shapes
are parallelograms.
Answer: If both pairs of opposite sides are the same length or if one pair
of opposite sides is congruent and parallel, the quadrilateral
is a parallelogram. If both pairs of opposite angles are
congruent or if the diagonals bisect each other, the
quadrilateral is a
parallelogram.
PROPERTIES OF PARALLELOGRAMS
Determine whether the quadrilateral is a parallelogram. Justify
your answer.
Answer: Each pair of opposite sides have the same measure. Therefore,
they are congruent. If both pairs of opposite sides of a
quadrilateral are congruent, the quadrilateral is a
parallelogram.
Determine whether the quadrilateral is a parallelogram. Justify
your answer.
Answer: One pair of opposite sides is parallel and has the same measure,
which means these sides are congruent. If one pair of
opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram.
FIND MEASURES
Find x so that the quadrilateral is a parallelogram.
A
B
D
C
Opposite sides of a parallelogram are congruent.
Substitution
Distributive Property
Subtract 3x from each side.
Add 1 to each side.
Answer: When x is 7, ABCD is a parallelogram.
Find y so that the quadrilateral is a parallelogram.
D
G
E
F
Opposite angles of a parallelogram are congruent.
Substitution
Subtract 6y from each side.
Subtract 28 from each side.
Divide each side by –1.
Answer: DEFG is a parallelogram when y is 14.
Find m and n so that each quadrilateral is a parallelogram.
a.
b.
Answer:
Answer:
USE SLOPE AND DISTANCE
COORDINATE GEOMETRY Determine whether the figure with
vertices A(–3, 0), B(–1, 3), C(3, 2), and
D(1, –1) is a parallelogram. Use the Slope Formula.
If the opposite sides of a quadrilateral are parallel, then it is a
parallelogram.
Answer: Since opposite sides have the same slope,
Therefore, ABCD is a
parallelogram by definition.
COORDINATE GEOMETRY Determine whether the figure with
vertices P(–3, –1), Q(–1, 3), R(3, 1), and
S(1, –3) is a parallelogram. Use the Distance and Slope Formulas.
First use the Distance Formula to determine whether the opposite sides
are congruent.
Next, use the Slope Formula to determine whether
and have the same slope, so they are parallel.
Answer: Since one pair of opposite sides is congruent and parallel, PQRS
is a parallelogram.
Determine whether the figure with the given vertices is a
parallelogram. Use the method indicated.
a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1);
Slope Formula
Answer: The slopes of
and the slopes
of
Therefore,
Since opposite sides are parallel,
ABCD is a parallelogram.
Determine whether the figure with the given vertices is a
parallelogram. Use the method indicated.
Distance and
Slope Formulas
b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);
Answer:
Since the slopes
of
Since one
pair of opposite sides is congruent and parallel, LMNO is a
parallelogram.
HOMEWORK:
P. 769, SECTIONS 8.2 AND 8.3- ALL