The document discusses properties of trapezoids and kites. It defines key terms like trapezoid, bases, legs, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. Theorems regarding isosceles trapezoids and kites are presented. Two examples apply the theorems to solve problems involving finding missing angle measures and side lengths of trapezoids.
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4. Vocabulary
1.Trapezoid: A quadrilateral with only one pair of
parallel sides
2. Bases:
3. Legs of a Trapezoid:
4. Base Angles:
5. Isosceles Trapezoid:
Wednesday, April 25, 2012
5. Vocabulary
1.Trapezoid: A quadrilateral with only one pair of
parallel sides
2. Bases: The parallel sides of a trapezoid
3. Legs of a Trapezoid:
4. Base Angles:
5. Isosceles Trapezoid:
Wednesday, April 25, 2012
6. Vocabulary
1.Trapezoid: A quadrilateral with only one pair of
parallel sides
2. Bases: The parallel sides of a trapezoid
3. Legs of a Trapezoid: The sides that are not parallel in a
trapezoid
4. Base Angles:
5. Isosceles Trapezoid:
Wednesday, April 25, 2012
7. Vocabulary
1.Trapezoid: A quadrilateral with only one pair of
parallel sides
2. Bases: The parallel sides of a trapezoid
3. Legs of a Trapezoid: The sides that are not parallel in a
trapezoid
4. Base Angles: The angles formed between a base and
one of the legs
5. Isosceles Trapezoid:
Wednesday, April 25, 2012
8. Vocabulary
1.Trapezoid: A quadrilateral with only one pair of
parallel sides
2. Bases: The parallel sides of a trapezoid
3. Legs of a Trapezoid: The sides that are not parallel in a
trapezoid
4. Base Angles: The angles formed between a base and
one of the legs
5. Isosceles Trapezoid: A trapezoid that has congruent
legs
Wednesday, April 25, 2012
10. Vocabulary
6. Midsegment of a Trapezoid: The segment that
connects the midpoints of the legs of a trapezoid
7. Kite:
Wednesday, April 25, 2012
11. Vocabulary
6. Midsegment of a Trapezoid: The segment that
connects the midpoints of the legs of a trapezoid
7. Kite: A quadrilateral with exactly two pairs of
consecutive congruent sides; Opposite sides are not
parallel or congruent
Wednesday, April 25, 2012
14. Theorems
Isosceles Trapezoid
6.21: If a trapezoid is isosceles, then each pair of base
angles is congruent
6.22: If a trapezoid has one pair of congruent base
angles, then it is isosceles
6.23:
Wednesday, April 25, 2012
15. Theorems
Isosceles Trapezoid
6.21: If a trapezoid is isosceles, then each pair of base
angles is congruent
6.22: If a trapezoid has one pair of congruent base
angles, then it is isosceles
6.23: A trapezoid is isosceles IFF its diagonals are
congruent
Wednesday, April 25, 2012
17. Theorems
6.24 - Trapezoid Midsegment Theorem: The midsegment
of a trapezoid is parallel to each base and its measure
is half of the sum of the lengths of the two bases
Kites
6.25:
6.26:
Wednesday, April 25, 2012
18. Theorems
6.24 - Trapezoid Midsegment Theorem: The midsegment
of a trapezoid is parallel to each base and its measure
is half of the sum of the lengths of the two bases
Kites
6.25: If a quadrilateral is a kite, then its diagonals are
perpendicular
6.26:
Wednesday, April 25, 2012
19. Theorems
6.24 - Trapezoid Midsegment Theorem: The midsegment
of a trapezoid is parallel to each base and its measure
is half of the sum of the lengths of the two bases
Kites
6.25: If a quadrilateral is a kite, then its diagonals are
perpendicular
6.26: If a quadrilateral is a kite, then exactly one pair of
opposite angles is congruent
Wednesday, April 25, 2012
20. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
Wednesday, April 25, 2012
21. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
Wednesday, April 25, 2012
22. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
360 − 2(130)
Wednesday, April 25, 2012
23. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
360 − 2(130) = 100
Wednesday, April 25, 2012
24. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
360 − 2(130) = 100
100/2
Wednesday, April 25, 2012
25. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
360 − 2(130) = 100
100/2 = 50
Wednesday, April 25, 2012
26. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
a. m∠MJK
m∠JML = m∠KLM
360 − 2(130) = 100
100/2 = 50
m∠MJK = 50°
Wednesday, April 25, 2012
27. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
b. JL
Wednesday, April 25, 2012
28. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
b. JL
JN = KN
Wednesday, April 25, 2012
29. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
b. JL
JN = KN
JL = JN + LN
Wednesday, April 25, 2012
30. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
b. JL
JN = KN
JL = JN + LN
JL = 6.7 + 3.6
Wednesday, April 25, 2012
31. Example 1
Each side of a basket is an isosceles trapezoid. If m∠JML =
130°, KN = 6.7 ft, and LN = 3.6 ft, find each measure.
b. JL
JN = KN
JL = JN + LN
JL = 6.7 + 3.6
JL = 10.3 units
Wednesday, April 25, 2012
32. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
Wednesday, April 25, 2012
33. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
Wednesday, April 25, 2012
34. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
Wednesday, April 25, 2012
35. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
Wednesday, April 25, 2012
36. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
C
Wednesday, April 25, 2012
37. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
C
D
Wednesday, April 25, 2012
38. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
C
D
Wednesday, April 25, 2012
39. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
C
D
We need AB to be parallel
with CD
Wednesday, April 25, 2012
40. Example 2
Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3),
and D(2, 4). Show that ABCD is a trapezoid and determine
whether it is an isosceles trapezoid.
x
y
A
B
C
D
We need AB to be parallel
with CD
CB ≅ ADAlso,
Wednesday, April 25, 2012
41. Example 2
A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4)
Wednesday, April 25, 2012
42. Example 2
A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4)
m(AB) =
−1−1
−3− 5
Wednesday, April 25, 2012
43. Example 2
A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4)
m(AB) =
−1−1
−3− 5
=
−2
−8
Wednesday, April 25, 2012
44. Example 2
A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4)
m(AB) =
−1−1
−3− 5
=
−2
−8
=
1
4
Wednesday, April 25, 2012
56. Example 2
A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4)
m(AB) =
−1−1
−3− 5
=
−2
−8
=
1
4
m(CD) =
4 − 3
2 − (−2)
=
1
4
AD = (5 − 2)2
+ (1− 4)2
= (3)2
+ (−3)2
= 9 + 9 = 18
BC = (−3+ 2)2
+ (−1− 3)2
= (−1)2
+ (−4)2
= 1+16
= 17
It is not isoscelesCB ≅ AD
Wednesday, April 25, 2012
57. Example 3
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x?
Wednesday, April 25, 2012
58. Example 3
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x?
MN =
KF + JG
2
Wednesday, April 25, 2012
59. Example 3
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x?
MN =
KF + JG
2
30 =
20 + JG
2
Wednesday, April 25, 2012
60. Example 3
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x?
MN =
KF + JG
2
30 =
20 + JG
2
60 = 20 + JG
Wednesday, April 25, 2012
61. Example 3
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x?
MN =
KF + JG
2
30 =
20 + JG
2
60 = 20 + JG
40 = JG
Wednesday, April 25, 2012
75. Problem Set
p. 440 #9-27 odd, 35-43 odd, 49, 65, 75, 77
“Do what you love, love what you do, leave the world a
better place and don't pick your nose.” - Jeff Mallett
Wednesday, April 25, 2012