Transcript 8.6 Trapezoids and Kites

Trapezoids
Section 6.5
Objectives
• Use properties of trapezoids.
Key Vocabulary
•
•
•
•
•
•
Trapezoid
Bases of a trapezoid
Legs of a trapezoid
Base angles of a trapezoid
Isosceles trapezoid
Midsegment of a trapezoid
Theorems
• 6.12 & 6.13 Isosceles Trapezoid Theorems
Trapezoids
What makes a quadrilateral a trapezoid?
Definition: Trapezoid
A trapezoid is a
quadrilateral with
exactly one pair of
parallel opposite
sides.
Trapezoid
• A trapezoid is a quadrilateral
with exactly one pair of parallel
sides.
• Each of the parallel sides is
called a base.
• The nonparallel sides are called
legs.
• Base angles of a trapezoid are
two consecutive angles whose
common side is a base.
Isosceles Trapezoid
• If the legs of a trapezoid are congruent, the
trapezoid is an isosceles trapezoid.
• The following theorems state the properties of
an isosceles trapezoid.
Isosceles Trapezoid Theorem 6.12
• If a trapezoid is isosceles, then each pair of
base angles is congruent.
Isosceles Trapezoid Theorem 6.13
• If a trapezoid has a pair
of congruent base
angles, then it is an
isosceles trapezoid.
• If ∠D≅∠C, then
trapezoid ABCD is an D
isosceles.
A
B
C
Isosceles Trapezoids
If the legs of a trapezoid are ≅, then it is called an
isosceles trapezoid.
Theorem 6.12: Both pairs of base s of an isosceles
trapezoid are ≅.
(A ≅ D and B≅ C)
Theorem 6.13: If one pair of base s are ≅, then it
is an isosceles trapezoid.
(If ∠B≅∠C, then trapezoid ABCD is isosceles)
Example 1:
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.
Your Turn:
The sides of a picture frame appear to be two adjacent
trapezoids. Determine if they are isosceles trapezoids.
Answer: yes
Example 2
A. BASKET Each side of the basket shown is an isosceles
trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6
feet, find mMJK.
Example 2
Since JKLM is a trapezoid, JK║LM.
mJML + mMJK = 180
mJML + 130 = 180
mJML = 50
Answer:
mJML = 50
Consecutive Interior
Angles Theorem
Substitution
Subtract 130 from each
side.
Your Turn:
A. Each side of the basket shown is an isosceles trapezoid.
If mFGH = 124, FI = 9.8 feet, and
IG = 4.3 feet, find mEFG.
A. 124
B. 62
C. 56
D. 112
Example 3
PQRS is an isosceles trapezoid. Find the
missing angle measures.
SOLUTION
1. PQRS is an isosceles trapezoid and R and S are a
pair of base angles. So, mR = mS = 50°.
2. Because S and P are same-side interior angles
formed by parallel lines, they are supplementary. So,
mP = 180° – 50° = 130°.
3. Because Q and P are a pair of base angles of an
isosceles trapezoid, mQ = mP = 130°.
Your Turn:
ABCD is an isosceles trapezoid. Find the missing angle
measures.
1.
ANSWER
mA = 80°;
mB = 80°;
mC = 100°
2.
ANSWER
mA = 110°;
mB = 110°;
mD = 70°
3.
ANSWER
mB = 75°;
mC = 105°;
mD = 105°
Trapezoid Midsegment
A midsegment of a
trapezoid is a
segment that
connects the
midpoints of the legs
of a trapezoids.
Medians of Trapezoids
 The segment that joins the midpoints of the legs of a
trapezoid is called the median (MN). It is also referred to as
the midsegment.
median
BC || AD
Trapezoid Midsegment Properties
• The midsegment of a
trapezoid is parallel to
each base and its
length is one half the
sum of the lengths of
the bases.
If
Review: Trapezoids
Trapezoid Characteristics
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
(mA + mC = 180, mB + mD = 180)
Midsegment is parallel to bases
Midsegment = ½ (base + base)
½(AB + CD)
A
leg
midpoint
base
midsegment
C
B
leg
midpoint
D
base
A
B
Isosceles Trapezoid Characteristics
Legs are congruent (AC  BD)
Base angle pairs congruent (A  B, C  D)
M
C
D
Example 4
In the figure, MN is the midsegment of trapezoid FGJK.
What is the value of x.
Example 4
Trapezoid Midsegment
Theorem
Substitution
Multiply each side by 2.
Subtract 20 from each side.
Answer:
x = 40
Your Turn:
WXYZ is an isosceles trapezoid with median
Find XY if JK = 18 and WZ = 25.
A. XY = 32
B. XY = 25
C. XY = 21.5
D. XY = 11
Example 5
Find the length of the midsegment DG of
trapezoid CEFH.
SOLUTION
Use the formula for the midsegment of a trapezoid.
1
Formula for midsegment of a trapezoid
DG = (EF + CH)
2
1
Substitute 8 for EF and 20 for CH.
= (8 + 20)
2
1
Add.
= (28)
2
= 14
ANSWER
Multiply.
The length of the midsegment DG is 14.
Your Turn:
Find the length of the midsegment MN of the trapezoid.
1.
ANSWER
11
ANSWER
8
ANSWER
21
2.
3.
Example 6:
DEFG is an isosceles trapezoid with median
Find m∠1, m∠2, m∠3, and m∠4. if
and
Because this
is an isosceles trapezoid,
Example 6:
Consecutive Interior Angles Theorem
Substitution
Combine like terms.
Divide each side by 9.
Answer:
Because
Your Turn:
WXYZ is an isosceles trapezoid
with median
Answer:
Because
REVIEW TRAPEZOIDS
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Rectangles
Rhombi
Squares
Trapezoids
Isosceles
Trapezoids
ONE PAIR OF PARALLEL
SIDES
Leg angle 1
base
leg
leg
base
Leg angle 2
Leg angles are supplementary
Base (b1)
Midsegment
=½ (b1 + b2)
Base (b2)
Midsegment is ½ the sum of
the lengths of the bases
base
leg
leg
base
Base angle 1
Base angle 2
Isosceles: Base angles are
congruent
Joke Time
• Why does a room full of married people look
so empty?
• There's not a Single person in it...
• What do you find in a clean nose?
• Fingerprints!
Assignment
Pg. 334 - 336 #1 – 25 odd, 35, 37