Download 8.5 Use Properties of Trapezoids and Kites

8.5
Use Properties of Trapezoids
and Kites
Goal
Your Notes
p Use properties of trapezoids and kites.
VOCABULARY
Trapezoid
Bases of a trapezoid
Base angles of a trapezoid
Legs of a trapezoid
Isosceles trapezoid
Midsegment of a trapezoid
Kite
Copyright © Holt McDougal. All rights reserved.
Lesson 8.5 • Geometry Notetaking Guide
221
8.5
Use Properties of Trapezoids
and Kites
Goal
Your Notes
p Use properties of trapezoids and kites.
VOCABULARY
Trapezoid A trapezoid is a quadrilateral with exactly
one pair of parallel sides.
Bases of a trapezoid The parallel sides of a
trapezoid are the bases.
Base angles of a trapezoid A trapezoid has two pairs
of base angles. Each pair shares a base as a side.
Legs of a trapezoid The nonparallel sides of a
trapezoid are the legs.
Isosceles trapezoid An isosceles trapezoid is a
trapezoid in which the legs are congruent.
Midsegment of a trapezoid The midsegment of
a trapezoid is the segment that connects the
midpoints of its legs.
Kite A kite is a quadrilateral that has two pairs of
consecutive congruent sides, but opposite sides
are not congruent.
Copyright © Holt McDougal. All rights reserved.
Lesson 8.5 • Geometry Notetaking Guide
221
Your Notes
Example 1
Use a coordinate plane
Show that CDEF is a trapezoid.
y
D(1, 3)
Solution
Compare the slopes of opposite sides.
}
Slope of DE 5
5
E(4, 4)
F(6, 2)
1
C(0, 0)
x
4
}
Slope of CF 5
5
5
}
}
}
The slopes of DE and CF are the same, so DE
}
Slope of EF 5
5
}
CF.
5
}
Slope of CD 5
5
5
}
}
}
The slopes of EF and CD are not the same, so EF is
}
to CD.
Because quadrilateral CDEF has exactly one pair of
, it is a trapezoid.
THEOREM 8.14
B
If a trapezoid is isosceles, then each
pair of base angles is
.
C
A
D
If trapezoid ABCD is isosceles, then
∠A > ∠
and ∠
> ∠C.
THEOREM 8.15
If a trapezoid has a pair of congruent
, then it is an isosceles
trapezoid.
B
C
A
D
If ∠A > ∠D (or if ∠B > ∠C), then trapezoid
ABCD is isosceles.
THEOREM 8.16
B
A trapezoid is isosceles if and only
if its diagonals are
.
Trapezoid ABCD is isosceles if and only if
222
Lesson 8.5 • Geometry Notetaking Guide
C
A
D
>
.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 1
Use a coordinate plane
Show that CDEF is a trapezoid.
y
D(1, 3)
Solution
Compare the slopes of opposite sides.
423
1
}
Slope of DE 5 } 5 }
421
E(4, 4)
F(6, 2)
1
C(0, 0)
x
4
3
220
}
2
Slope of CF 5 } 5 } 5
620
6
}
}
The slopes of DE and CF are the
1
3
}
}
}
same, so DE i CF.
224
22
}
Slope of EF 5 } 5 } 5 21
624
2
320
3
}
Slope of CD 5 } 5 } 5 3
1
120
}
}
}
The slopes of EF and CD are not the same, so EF is
}
not parallel to CD.
Because quadrilateral CDEF has exactly one pair of
parallel sides , it is a trapezoid.
THEOREM 8.14
B
If a trapezoid is isosceles, then each
pair of base angles is congruent .
C
A
D
If trapezoid ABCD is isosceles, then
∠A > ∠ D and ∠ B > ∠C.
THEOREM 8.15
If a trapezoid has a pair of congruent
base angles , then it is an isosceles
trapezoid.
B
C
A
D
If ∠A > ∠D (or if ∠B > ∠C), then trapezoid
ABCD is isosceles.
THEOREM 8.16
B
A trapezoid is isosceles if and only
if its diagonals are congruent .
A
}
C
}
D
Trapezoid ABCD is isosceles if and only if AC > BD .
222
Lesson 8.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 2
Use properties of isosceles trapezoids
Kitchen A shelf fitting into a
cupboard in the corner of a
kitchen is an isosceles trapezoid.
Find m∠N, m∠L, and m∠M.
L
M
508
K
N
Solution
, so
Step 1 Find m∠N. KLMN is an
∠N and ∠
are congruent base angles, and
m∠N 5 m∠
5
.
Step 2 Find m∠L. Because ∠K and ∠L are consecutive
interior angles formed by @##$
KL intersecting two
parallel lines, they are
. So,
m∠L 5
2
5
.
are a pair
Step 3 Find m∠M. Because ∠M and ∠
of base angles, they are congruent, and
5
.
m∠M 5 m∠
So, m∠N 5
, m∠L 5
, and m∠M 5
.
Checkpoint Complete the following exercises.
1. In Example 1, suppose the coordinates of point E are
(7, 5). What type of quadrilateral is CDEF ? Explain.
2. Find m∠C, m∠A, and m∠D
in the trapezoid shown.
A
D
1358
B
Copyright © Holt McDougal. All rights reserved.
C
Lesson 8.5 • Geometry Notetaking Guide
223
Your Notes
Example 2
Use properties of isosceles trapezoids
Kitchen A shelf fitting into a
cupboard in the corner of a
kitchen is an isosceles trapezoid.
Find m∠N, m∠L, and m∠M.
L
M
508
K
N
Solution
Step 1 Find m∠N. KLMN is an isosceles trapezoid , so
∠N and ∠ K are congruent base angles, and
m∠N 5 m∠ K 5 508 .
Step 2 Find m∠L. Because ∠K and ∠L are consecutive
interior angles formed by @##$
KL intersecting two
parallel lines, they are supplementary . So,
m∠L 5 1808 2 508 5 1308 .
Step 3 Find m∠M. Because ∠M and ∠ L are a pair
of base angles, they are congruent, and
m∠M 5 m∠ L 5 1308 .
So, m∠N 5 508 , m∠L 5 1308 , and m∠M 5 1308 .
Checkpoint Complete the following exercises.
1. In Example 1, suppose the coordinates of point E are
(7, 5). What type of quadrilateral is CDEF ? Explain.
Parallelogram; opposite pairs of sides are
parallel.
2. Find m∠C, m∠A, and m∠D
in the trapezoid shown.
m∠C 5 1358, m∠A 5 458,
m∠D 5 458
Copyright © Holt McDougal. All rights reserved.
A
D
1358
B
C
Lesson 8.5 • Geometry Notetaking Guide
223
Your Notes
THEOREM 8.17: MIDSEGMENT THEOREM FOR
TRAPEZOIDS
A
The midsegment of a trapezoid is
parallel to each base and its length
M
is one half the sum of the lengths
D
of the bases.
}
If MN is the midsegment of trapezoid ABCD, then
}
}
MN i
, MN i
, and MN 5
(
1
B
N
C
).
Use the midsegment of a trapezoid
Example 3
}
In the diagram, MN is the midsegment
of trapezoid PQRS. Find MN.
(
5
(
1
1
5
The length MN is
)
)
Q
M
Solution
Use Theorem 8.17 to find MN.
MN 5
16 in.
P
N
S
9 in.
R
Apply Theorem 8.17.
Substitute
for SR.
for PQ and
Simplify.
inches.
Checkpoint Complete the following exercise.
3. Find MN in the trapezoid at
the right.
P
M
Q
12 ft
R
30 ft
N
S
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Lesson 8.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 8.17: MIDSEGMENT THEOREM FOR
TRAPEZOIDS
A
B
The midsegment of a trapezoid is
parallel to each base and its length
M
N
is one half the sum of the lengths
D
C
of the bases.
}
If MN is the midsegment of trapezoid ABCD, then
1
} } } }
MN i AB , MN i DC , and MN 5 } ( AB 1 CD ).
2
Example 3
Use the midsegment of a trapezoid
}
In the diagram, MN is the midsegment
of trapezoid PQRS. Find MN.
2
1
5 } ( 16 1 9 )
2
5 12.5
Q
M
Solution
Use Theorem 8.17 to find MN.
1
MN 5 } ( PQ 1 SR )
16 in.
P
N
S
9 in.
R
Apply Theorem 8.17.
Substitute 16 for PQ and
9 for SR.
Simplify.
The length MN is 12.5 inches.
Checkpoint Complete the following exercise.
3. Find MN in the trapezoid at
the right.
MN 5 21 ft
P
M
Q
12 ft
R
30 ft
N
S
224
Lesson 8.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 8.18
If a quadrilateral is a kite, then its
diagonals are
.
C
B
D
If quadrilateral ABCD is a kite,
then
⊥
.
A
THEOREM 8.19
C
If a quadrilateral is a kite, then
exactly one pair of opposite angles
are congruent.
B
D
A
}
}
If quadrilateral ABCD is a kite and BC > BA,
then ∠A
∠C and ∠B
∠D.
Example 4
Apply Theorem 8.19
Find m∠T in the kite shown at the right.
Q
R
708
Solution
By Theorem 8.19, QRST has exactly one
pair of
opposite angles.
T
Because ∠Q À ∠S, ∠
and ∠T must be
congruent. So, m∠
5 m∠T. Write and solve
an equation to find m∠T.
S
m∠T 1 m∠R 1
1
5
Corollary to
Theorem 8.1
m∠T 1 m∠T 1
1
5
Substitute m∠T
for m∠R.
5
Combine like
terms.
(m∠T ) 1
m∠T 5
Homework
888
Solve for m∠T.
Checkpoint Complete the following exercise.
4. Find m∠G in the kite shown
at the right.
G
758
J
Copyright © Holt McDougal. All rights reserved.
H
858
Lesson 8.5 • Geometry Notetaking Guide
I
225
Your Notes
THEOREM 8.18
If a quadrilateral is a kite, then its
diagonals are perpendicular .
C
B
D
If quadrilateral ABCD is a kite,
}
}
then AC ⊥ BD .
A
THEOREM 8.19
C
If a quadrilateral is a kite, then
exactly one pair of opposite angles
are congruent.
B
D
A
}
}
If quadrilateral ABCD is a kite and BC > BA,
then ∠A > ∠C and ∠B À ∠D.
Example 4
Apply Theorem 8.19
Find m∠T in the kite shown at the right.
Q
R
708
Solution
By Theorem 8.19, QRST has exactly one
pair of congruent opposite angles.
T
Because ∠Q À ∠S, ∠ R and ∠T must be
congruent. So, m∠ R 5 m∠T. Write and solve
an equation to find m∠T.
S
m∠T 1 m∠R 1 708 1 888 5 3608
Corollary to
Theorem 8.1
m∠T 1 m∠T 1 708 1 888 5 3608
Substitute m∠T
for m∠R.
2 (m∠T ) 1 1588 5 3608
Combine like
terms.
m∠T 5 1018
Homework
888
Solve for m∠T.
Checkpoint Complete the following exercise.
4. Find m∠G in the kite shown
at the right.
G
m∠G 5 1008
758
J
Copyright © Holt McDougal. All rights reserved.
H
858
Lesson 8.5 • Geometry Notetaking Guide
I
225