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Lesson 8-1 Angles of Polygons
Lesson 8-2 Parallelograms
Lesson 8-3 Tests for Parallelograms
Lesson 8-4 Rectangles
Lesson 8-5 Rhombi and Squares
Lesson 8-6 Trapezoids
Lesson 8-7 Coordinate Proof with Quadrilaterals
Example 1 Interior Angles of Regular Polygons
Example 2 Sides of a Polygon
Example 3 Interior Angles
Example 4 Exterior Angles
ARCHITECTURE A mall
is designed so that five
walkways meet at a food
court that is in the shape
of a regular pentagon.
Find the sum of measures
of the interior angles of
the pentagon.
Since a pentagon is a convex polygon, we can use the
Angle Sum Theorem.
Interior Angle Sum Theorem
Simplify.
Answer: The sum of the measures of the angles is 540.
A decorative window is designed to have the shape of
a regular octagon. Find the sum of the measures of the
interior angles of the octagon.
Answer: 1080
The measure of an interior angle of a regular polygon
is 135. Find the number of sides in the polygon.
Use the Interior Angle Sum Theorem to write an equation
to solve for n, the number of sides.
Interior Angle Sum Theorem
Distributive Property
Subtract 135n from each side.
Add 360 to each side.
Divide each side by 45.
Answer: The polygon has 8 sides.
The measure of an interior angle of a regular polygon
is 144. Find the number of sides in the polygon.
Answer: The polygon has 10 sides.
Find the measure of each interior angle.
Since
the sum of the measures of the interior
angles is
Write an equation to express
the sum of the measures of the interior angles
of the polygon.
Sum of measures of
angles
Substitution
Combine like terms.
Subtract 8 from each
side.
Divide each side by
32.
Use the value of x to find the measure of each angle.
Answer:
Find the measure of each interior angle.
Answer:
Find the measures of an exterior angle and an interior
angle of convex regular nonagon ABCDEFGHJ.
At each vertex, extend a side to form one exterior angle.
The sum of the measures of the exterior angles is 360. A
convex regular nonagon has 9 congruent exterior angles.
Divide each side by 9.
Answer: The measure of each exterior angle is 40. Since
each exterior angle and its corresponding
interior angle form a linear pair, the measure of
the interior angle is 180 – 40 or 140.
Find the measures of an exterior angle and an interior
angle of convex regular hexagon ABCDEF.
Answer: 60; 120
Example 1 Proof of Theorem 8.4
Example 2 Properties of Parallelograms
Example 3 Diagonals of a Parallelogram
Prove that if a parallelogram has two consecutive
sides congruent, it has four sides congruent.
Given:
Prove:
Proof:
Statements
1.
2.
3.
4.
Reasons
1. Given
2. Given
3. Opposite sides of a
parallelogram are .
4. Transitive Property
Prove that if
Given:
Prove:
and
and
are the diagonals of
,
Proof:
Statements
Reasons
1.
1. Given
2.
2. Opposite sides of a
parallelogram are congruent.
3.
3. If 2 lines are cut by a transversal,
alternate interior s are .
4.
4. Angle-Side-Angle
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:
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
Read the Test Item
Since the diagonals of a parallelogram bisect each other,
the intersection point is the midpoint of
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
Example 1 Write a Proof
Example 2 Properties of Parallelograms
Example 3 Properties of Parallelograms
Example 4 Find Measures
Example 5 Use Slope and Distance
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.
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.
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 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:
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.
b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2); Distance and
Slope Formulas
Answer:
Since the
slopes of
Since one pair of opposite sides is congruent
and parallel, LMNO is a parallelogram.
Example 1 Diagonals of a Rectangle
Example 2 Angles of a Rectangle
Example 3 Diagonals of a Parallelogram
Example 4 Rectangle on a Coordinate Plane
Quadrilateral RSTU is a rectangle. If
find x.
and
The diagonals of a rectangle are congruent,
Definition of congruent segments
Substitution
Subtract 6x from each side.
Add 4 to each side.
Answer: 8
Quadrilateral EFGH is a rectangle. If
find x.
Answer: 5
and
Quadrilateral LMNP is a rectangle. Find x.
Angle Addition Theorem
Substitution
Simplify.
Subtract 10 from each side.
Divide each side by 8.
Answer: 10
Quadrilateral LMNP is a rectangle. Find y.
Since a rectangle is a parallelogram, opposite sides are
parallel. So, alternate interior angles are congruent.
Alternate Interior Angles Theorem
Substitution
Simplify.
Subtract 2 from each side.
Divide each side by 6.
Answer: 5
Quadrilateral EFGH is a rectangle.
a. Find x.
b. Find y.
Answer: 7
Answer: 11
Kyle is building a barn for his horse. He measures the
diagonals of the door opening to make sure that they
bisect each other and they are congruent. How does
he know that the corners are
angles?
Answer: We know that
A parallelogram with
congruent diagonals is a rectangle. Therefore,
the corners are
angles.
Max is building a swimming
pool in his backyard. He
measures the length and
width of the pool so that
opposite sides are parallel. He
also measures the diagonals
of the pool to make sure that
they are congruent. How does
he know that the measure of
each corner is 90?
Answer: Since opposite sides are parallel, we know that
RSTU is a parallelogram. We know that
.
A parallelogram with congruent diagonals is a
rectangle. Therefore, the corners are
Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0),
and D(–1, –2). Determine whether ABCD is a rectangle
using the Slope Formula.
Method 1: Use the Slope Formula,
to see if
consecutive sides are perpendicular.
quadrilateral ABCD is a
parallelogram. The product of the slopes of consecutive
sides is –1. This means that
Answer: The perpendicular segments create four right
angles. Therefore, by definition ABCD is a
rectangle.
Method 2: Use the Distance Formula,
to determine whether
opposite sides are congruent.
Since each pair of opposite sides of the quadrilateral have
the same measure, they are congruent. Quadrilateral
ABCD is a parallelogram.
Find the length of the diagonals.
The length of each diagonal is
Answer: Since the diagonals are congruent, ABCD is a
rectangle.
Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3),
Y(3, 1), and Z(2, –1). Determine whether WXYZ is a
rectangle using the Distance Formula.
Answer:
we
can conclude that opposite sides of the
quadrilateral are congruent. Therefore, WXYZ is
a parallelogram. Diagonals WY and XZ each
have a length of 5. Since the diagonals are
congruent, WXYZ is a rectangle by Theorem
8.14.
Example 1 Proof of Theorem 8.15
Example 2 Measures of a Rhombus
Example 3 Squares
Example 4 Diagonals of a Square
Given: BCDE is a rhombus,
and
Prove:
D
Proof: Because opposite angles of a rhombus are
congruent and the diagonals of a rhombus bisect each
other,
By substitution,
by the Reflexive Property and it is given that
Therefore,
by SAS.
Given: ACDF is a rhombus;
Prove:
Proof: Since ACDF is a rhombus, diagonals
bisect each other and are perpendicular to each other.
Therefore,
are both right
angles. By definition of right angles,
which means that
by
definition of congruent angles. It is given that
so
since alternate interior angles are
congruent when parallel lines are cut by a transversal.
by ASA.
Use rhombus LMNP to find the value of y if
N
The diagonals of a rhombus are
perpendicular.
Substitution
Add 54 to each side.
Take the square root of each side.
Answer: The value of y can be 12 or –12.
Use rhombus LMNP to find
if
N
Opposite angles are congruent.
Substitution
The diagonals of a rhombus bisect the angles.
Answer:
Use rhombus ABCD and
the given information to
find the value of each
variable.
a.
Answer: 8 or –8
b.
Answer:
Determine whether parallelogram ABCD is a rhombus,
a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2),
and D(2, –2). List all that apply. Explain.
Explore Plot the vertices on a coordinate plane.
Plan
If the diagonals are perpendicular, then ABCD
is either a rhombus or a square. The diagonals
of a rectangle are congruent. If the diagonals are
congruent and perpendicular, then ABCD is a
square.
Solve Use the Distance Formula to compare the lengths
of the diagonals.
Use slope to determine whether the diagonals are
perpendicular.
Since the slope of
is the negative reciprocal of the
slope of
the diagonals are perpendicular. The lengths
of
and
are the same so the diagonals
are congruent. ABCD is a rhombus, a rectangle, and
a square.
Examine The diagonals are congruent and perpendicular
so ABCD must be a square. You can verify that
ABCD is a rhombus by finding AB, BC, CD, AD.
Then see if two consecutive segments are
perpendicular.
Answer: ABCD is a rhombus, a rectangle, and a square.
Determine whether parallelogram EFGH is a rhombus,
a rectangle, or a square for E(0, –2), F(–3, 0), G(–1, 3),
and H(2, 1). List all that apply. Explain.
Answer:
and
slope of
slope of
Since the slope of
is the
negative reciprocal of the slope of
, the
diagonals are perpendicular. The lengths of
and
are the same.
A square table has four legs
that are 2 feet apart. The table
is placed over an umbrella
stand so that the hole in the
center of the table lines up
with the hole in the stand.
How far away from a leg is
the center of the hole?
Let ABCD be the square formed by the legs of the table.
Since a square is a parallelogram, the diagonals bisect
each other. Since the umbrella stand is placed so that its
hole lines up with the hole in the table, the center of the
umbrella pole is at point E, the point where the diagonals
intersect. Use the Pythagorean Theorem to
find the length of a diagonal.
The distance from the center of the pole to a leg is equal
to the length of
Answer: The center of the pole is about 1.4 feet from a leg
of a table.
Kayla has a garden whose length and width are each
25 feet. If she places a fountain exactly in the center of
the garden, how far is the center of the fountain from
one of the corners of the garden?
Answer: about 17.7 feet
Example 1 Proof of Theorem 8.19
Example 2 Identify Isosceles Trapezoids
Example 3 Identify Trapezoids
Example 4 Median of a Trapezoid
Write a flow proof.
Given: KLMN is an isosceles
trapezoid.
Prove:
Proof:
Write a flow proof.
Given: ABCD is an isosceles trapezoid.
Prove:
Proof:
The top of this work station appears to be two adjacent
trapezoids. Determine if they are isosceles trapezoids.
Each pair of base angles is congruent, so the legs are the
same length.
Answer: Both trapezoids are isosceles.
The sides of a picture frame appear to be two adjacent
trapezoids. Determine if they are isosceles trapezoids.
Answer: yes
ABCD is a quadrilateral with vertices A(5, 1), B(–3, –1),
C(–2, 3), and D(2, 4). Verify that ABCD is a trapezoid.
A quadrilateral is a trapezoid if exactly one pair of opposite
sides are parallel. Use the Slope Formula.
slope of
slope of
slope of
slope of
Answer: Exactly one pair of opposite sides are parallel,
So, ABCD is a trapezoid.
ABCD is a quadrilateral with vertices A(5, 1), B(–3, 1),
C(–2, 3), and D(2, 4). Determine whether ABCD is an
isosceles trapezoid. Explain.
First use the Distance Formula to show that the
legs are congruent.
Answer: Since the legs are not congruent, ABCD is not
an isosceles trapezoid.
QRST is a quadrilateral with vertices Q(–3, –2), R(–2, 2),
S(1, 4), and T(6, 4).
a. Verify that QRST is a trapezoid.
Answer: Exactly one pair of opposite sides is parallel.
Therefore, QRST is a trapezoid.
b. Determine whether QRST is an isosceles trapezoid.
Explain.
Answer: Since the legs are not congruent, QRST
is not an isosceles trapezoid.
DEFG is an isosceles trapezoid with median
DG if
and
Find
Theorem 8.20
Substitution
Multiply each side by 2.
Subtract 20 from each side.
Answer:
DEFG is an isosceles trapezoid with median
Find
, and
if
and
Because this
is an isosceles trapezoid,
Consecutive Interior Angles Theorem
Substitution
Combine like terms.
Divide each side by 9.
Answer:
Because
WXYZ is an isosceles trapezoid
with median
a.
Answer:
b.
Answer:
Because
Example 1 Positioning a Square
Example 2 Find Missing Coordinates
Example 3 Coordinate Proof
Example 4 Properties of Quadrilaterals
Position and label a rectangle with sides a and b units
long on the coordinate plane.
Let A, B, C, and D be vertices of a rectangle with sides
a units long, and sides
b units
long.
Place the square with vertex A at the origin,
along the
positive x-axis, and
along the y-axis. Label the vertices
A, B, C, and D.
The y-coordinate of B is 0 because the vertex is on the
x-axis. Since the side length is a, the x-coordinate is a.
D is on the y-axis so the x-coordinate is 0. Since the side
length is b, the y-coordinate is b.
The x-coordinate of C is also a. The y-coordinate is
b because the side
is b units long.
Sample answer:
Position and label a parallelogram with sides a and b
units long on the coordinate plane.
Sample answer:
Name the missing coordinates for the isosceles
trapezoid.
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
and the
y-coordinate of D is
Answer:
Name the missing coordinates for the rhombus.
Answer:
Place a rhombus on the coordinate plane. Label the
midpoints of the sides M, N, P, and Q. Write a
coordinate proof to prove that MNPQ is a rectangle.
The first step is to position a
rhombus on the coordinate
plane so that the origin is the
midpoint of the diagonals and
the diagonals are on the axes,
as shown. Label the vertices
to make computations as
simple as possible.
Given: ABCD is a rhombus as labeled. M, N, P, Q are
midpoints.
Prove: MNPQ is a rectangle.
Proof:
By the Midpoint Formula, the coordinates of M, N, P, and Q
are as follows.
Find the slopes of
slope of
slope of
slope of
slope of
A segment with slope 0 is perpendicular to a segment
with undefined slope. Therefore, consecutive sides of
this quadrilateral are perpendicular. Since consecutive
sides are perpendicular, MNPQ is, by definition,
a rectangle.
Place an isosceles
trapezoid on the
coordinate plane.
Label the midpoints of
the sides M, N, P, and
Q. Write a coordinate
proof to prove that
MNPQ is a rhombus.
Given: ABCD is an isosceles trapezoid.
M, N, P, and Q are midpoints.
Prove: MNPQ is a rhombus.
Proof:
The coordinates of M are (–3a, b); the coordinates of N are
(0, 0); the coordinates of P are (3a, b); the coordinates of Q
are (0, 2b).
Since opposite sides have
equal slopes, opposite sides are parallel.
Since all four sides are congruent and
opposite sides are parallel, MNPQ is a rhombus.
Write a coordinate proof
to prove that the supports
of a platform lift are parallel.
Given: A(5, 0), B(10, 5),
C(5, 10), D(0, 5)
Prove:
Proof:
Since
parallel.
have the same slope, they are
Write a coordinate proof
to prove that the
crossbars of a child
safety gate are parallel.
Given: A(–3, 4), B(1, –4),
C(–1, 4), D(3, –4)
Prove:
Proof:
Since
have the same slope, they are parallel.
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