Download Lesson

Lesson 6-6
Trapezoids and Kites
5-Minute Check on Lesson 6-5
LMNO is a rhombus.
1. Find x
2. Find y
L
(8y – 6)°
(3x + 12)°
7
O
12
M
P
(5x – 2)°
QRST is a square.
N
3. Find n if mTQR = 8n + 8.
10.25
Q
4. Find w if QR = 5w + 4 and RS = 2(4w – 7).
5. Find QU if QS = 16t – 14 and QU = 6t + 11.
6. Standardized Test Practice:
not to a rhombus?
R
U
6
65
T
S
What property applies to a square, but
A
Opposite sides are congruent
C
Diagonals bisect each other
B
Opposite angles are congruent
D
All angles are right angles
Click the mouse button or press the
Space Bar to display the answers.
Objectives
• Recognize and apply the properties of
trapezoids
• Solve problems involving medians of
trapezoids
• Apply properties of kites
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
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Rectangles
Rhombi
Squares
Kites
Trapezoids
Isosceles
Trapezoids
Trapezoids
Trapezoid Characteristics
A
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
(mA + mC = 180, mB + mD = 180)
Median is parallel to bases
Median = ½ (base + base)
leg
midpoint
base
median
C
½(AB + CD)
B
leg
midpoint
D
base
A
B
Isosceles Trapezoid Characteristics
Legs are congruent (AC  BD)
Base angle pairs congruent (A  B, C  D)
Diagonals are congruent (AD  BC)
M
C
D
Kites Characteristics
A
Two pairs of consecutive sides congruent
AB  AD and CB  CD
Diagonals are perpendicular
AC  BD
D
B
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
C
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
Answer: mXYZ = 45
Simplify.
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 ΔMNR.
NR2 + MR2 = MN2
(6)2 + (8)2 = MN2
36 + 64 = MN2
100 = MN2
10 = MN
Pythagorean Theorem
Substitution
Simplify.
Add.
Take the square root of
each side.
Answer: Since MN  NP, MN = NP. By substitution, NP = 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.
Example 3
The sides of a picture frame appear to be two adjacent
trapezoids. Determine if they are isosceles trapezoids.
Answer: yes
Example 4a
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:
Example 4b
DEFG is an isosceles trapezoid with median
Find
, and
if
and
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°
Example 5
WXYZ is an isosceles trapezoid
with median
a.
Answer:
b.
Answer:
Because
Quadrilateral Characteristics Summary
Convex Quadrilaterals
Parallelograms
4 sided polygon
4 interior angles sum to 360
4 exterior angles sum to 360
Trapezoids
Bases Parallel
Opposite sides parallel and congruent
Legs are not Parallel
Opposite angles congruent
Leg angles are supplementary
Consecutive angles supplementary
Median is parallel to bases
2 congruent sides (consecutive)
Diagonals bisect each other
Median = ½ (base + base)
Diagonals perpendicular
Diagonals bisect opposite angles
One diagonal bisected
One pair of opposite angle congruent
Kites
Rectangles
Rhombi
Angles all 90°
Diagonals congruent
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Squares
Diagonals divide into 4 congruent triangles
Isosceles
Trapezoids
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
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