Trapezoids and Kites

1
Lesson 6-6

• Trapezoids and Kites

2
5-Minute Check on Lesson 6-5

• LMNO is a rhombus.
• Find x
• Find y
• QRST is a square.
• Find n if m?TQR 8n 8.
• Find w if QR 5w 4 and RS 2(4w 7).
• Find QU if QS 16t 14 and QU 6t 11.
• 6.
  What property applies to a square, but not to
  a rhombus?

Standardized Test Practice
A
C
Opposite sides are congruent
Diagonals bisect each other
B
D
Opposite angles are congruent
All angles are right angles
Click the mouse button or press the Space Bar to
display the answers.
3
Objectives

• Recognize and apply the properties of trapezoids
• Solve problems involving medians of trapezoids
• Apply properties of kites

4
Vocabulary

• Trapezoid a quadrilateral with only one pair of
  parallel sides
• Isosceles Trapezoid a trapezoid with both legs
  (non parallel sides) congruent
• Median a segment that joins the midpoints of
  the legs of a trapezoid

5
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Kites
Trapezoids
IsoscelesTrapezoids
Rhombi
Rectangles
Squares
6
Trapezoids
Trapezoid CharacteristicsBases Parallel Legs are
not Parallel Leg angles are supplementary (m?A
m?C 180, m?B m?D 180) Median is parallel
to basesMedian ½ (base base) ½(AB CD)
base
A
B
legmidpoint
legmidpoint
median
C
D
base
A
B
Isosceles Trapezoid CharacteristicsLegs are
congruent (AC ? BD) Base angle pairs congruent
(?A ? ?B, ?C ? ?D) Diagonals are congruent (AD ?
BC)
M
C
D
7
Kites Characteristics
Two pairs of consecutive sides congruent AB
? AD and CB ? CD Diagonals are perpendicular
AC ? BD Diagonals are angle bisectors
?BAC ? ?DAC and ?ABD ? ?CBD ?ADB ? ?BDC and
?BCA ? ?DCA Diagonal from noncongruent angles
bisects other diagonal diagonal BD is
cut in half Only one pair of opposite angles are
congruent (the pair of angles formed by the
non-congruent sides) ?ABC
? ?ADC
8
Example 1a
A. If WXYZ is a kite, find m?XYZ.
Since a kite only has one pair of congruent
angles, which are between the two non-congruent
sides, ?WXY ? ?WZY. So, ?WZY 121?.
m?W m?X m?Y m?Z 360 Polygon Interior
Angles Sum Theorem 73 121 m?Y
121 360 Substitution m?Y 45 Simplify.
Answer m?XYZ 45
9
Example 1b
B. If WXYZ is a kite, find NP.
Since the diagonals of a kite are perpendicular,
they divide MNPQ into four right triangles. Use
the Pythagorean Theorem to find MN, the length of
the hypotenuse of right ?MNR.
NR2 MR2 MN2 Pythagorean Theorem (6)2
(8)2 MN2 Substitution 36 64 MN2 Simplify.
100 MN2 Add. 10 MN Take the square root
of each side.
Answer Since MN ? NP, MN NP. By substitution,
NP 10
10
Example 2
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.
11
Example 3
The sides of a picture frame appear to be two
adjacent trapezoids. Determine if they are
isosceles trapezoids.
Answer yes
12
Example 4a
Theorem 8.20
Substitution
Multiply each side by 2.
Subtract 20 from each side.
13
Example 4b
Since EF // DG, ?1 and ?3 are supplementary

Because this is an isosceles
trapezoid, ?1 ? ?2 and ?3 ? ?4
Substitution
Combine like terms.
Divide each side by 9.
Answer If x 20, then m?1 65 and ?3 115.
Because ?1 ? ?2 and ?3 ? ?4, ?2 65 and ?4
115
14
Example 5
15
Quadrilateral Characteristics Summary
Convex Quadrilaterals
4 sided polygon 4 interior angles sum to 360 4
exterior angles sum to 360
Parallelograms
Trapezoids
Bases Parallel Legs are not Parallel Leg angles
are supplementary Median is parallel to
basesMedian ½ (base base)
Opposite sides parallel and congruent Opposite
angles congruent Consecutive angles
supplementary Diagonals bisect each other
Kites
2 congruent sides (consecutive) Diagonals
perpendicular Diagonals bisect opposite
angles One diagonal bisected One pair of opposite
angle congruent
Rectangles
Rhombi
IsoscelesTrapezoids
Angles all 90 Diagonals congruent
All sides congruent Diagonals perpendicular Diagon
als bisect opposite angles
Legs are congruent Base angle pairs congruent
Diagonals are congruent
Squares
Diagonals divide into 4 congruent triangles
16
Summary Homework

• Summary
• In an isosceles trapezoid, both pairs of base
  angles are congruent and the diagonals are
  congruent.
• The median of a trapezoid is parallel to the
  bases and its measure is one-half the sum of the
  measures of the bases
• Homework
• pg 440-42 1, 6, 7, 16-21, 41