Transcript Chapter 4 Congruent Triangles (page 116)

Lesson 5-5:
Trapezoids
(page 190)
Essential Question
How can the properties of quadrilaterals
be used to solve real life problems?
TRAPEZOID:
a quadrilateral with exactly one pair of parallel sides.
BASES of a TRAPEZOID : the
LEGS of a TRAPEZOID : the
parallel
sides.
non-parallel
sides.
base ➤
leg
leg
base
➤
Trapezoid
➤
leg
➤
base
base
leg
ISOSCELES TRAPEZOID: a trapezoid with
➤
➤
congruent
legs.
Theorem 5-18
Base angles of an isosceles trapezoid are
B
➤
X
Proof:
congruent .
For an outline of this proof see page 190.
Note that parallelograms are used in the proof.
➤
A
Y
MEDIAN of a TRAPEZOID: is the segment that joins
the
●
midpoints
of the legs.
●
●
Median of a Trapezoid
Review:
Median of a Triangle
Theorem 5-19
The median of a trapezoid:
(1) is parallel to the bases;
(2) has a length equal to the average of the base lengths.
Given: Trapezoid PQRS with median MN
S
R
Prove:
M
P
Proof:
N
Q
For an outline of this proof see page 191. Note congruent triangles are used in the proof.
Theorem 5-19
The median of a trapezoid:
(1) is parallel to the bases;
(2) has a length equal to the average of the base lengths.
Given: Trapezoid PQRS with median MN
S
Prove:
M
P
Proof:
➤
➤
➤
R
N
Q
For an outline of this proof see page 191. Note congruent triangles are used in the proof.
Theorem 5-19
The median of a trapezoid:
(1) is parallel to the bases;
(2) has a length equal to the average of the base lengths.
Given: Trapezoid PQRS with median MN
S
Prove:
M
P
Proof:
➤
➤
➤
R
N
Q
For an outline of this proof see page 191. Note congruent triangles are used in the proof.
Example #1: Given a trapezoid and its median, find the value of “x”.
6
x
12
Example #2: Given a trapezoid and its median, find the value of “x”.
7x-2
5x
x + 12
The QUADRILATERAL Hierarchy
Quadrilateral
Parallelogram
Trapezoid
Rhombus
Rectangle
Isosceles
Trapezoid
Square
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Diagonals are congruent
Both pairs off opposite
sides are congruent
At least one right angle
Both pairs off opposite
angles are congruent
Exactly one pair of opposite
sides are parallel
Diagonals perpendicular
Consecutive sides are
congruent
Consecutive angles are
congruent
Diagonals bisect each other
Diagonals bisect opposite
angles
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs off opposite
angles are congruent
X
Exactly one pair of opposite
sides are parallel
Diagonals perpendicular
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
Diagonals bisect opposite
angles
Rectangle
Rhombus
Square
Trapezoid
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs off opposite
angles are congruent
X
Rectangle
X
X
X
X
X
Exactly one pair of opposite
sides are parallel
Diagonals perpendicular
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
Diagonals bisect opposite
angles
X
X
Rhombus
Square
Trapezoid
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs off opposite
angles are congruent
X
Rectangle
X
X
X
X
X
Rhombus
X
X
X
Exactly one pair of opposite
sides are parallel
Diagonals perpendicular
X
X
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
Diagonals bisect opposite
angles
X
X
supp
X
X
Square
Trapezoid
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs off opposite
angles are congruent
X
Rectangle
X
X
X
X
X
Rhombus
X
X
X
Square
X
X
X
X
X
Exactly one pair of opposite
sides are parallel
Diagonals perpendicular
X
X
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
Diagonals bisect opposite
angles
X
X
supp
X
X
X
X
X
X
X
Trapezoid
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs of opposite
angles are congruent
X
Rectangle
X
X
X
X
X
Rhombus
X
X
X
Square
X
X
X
X
X
Exactly one pair of opposite
sides are parallel
X
Diagonals perpendicular
X
X
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
Diagonals bisect opposite
angles
Trapezoid
X
X
supp
X
X
X
X
X
X
X
Isosceles
Trapezoid
CHARACTERISTICS
Both pairs off opposite
sides are parallel
Parallelogram
X
Diagonals are congruent
Both pairs off opposite
sides are congruent
X
At least one right angle
Both pairs off opposite
angles are congruent
X
Rectangle
X
X
X
X
X
Rhombus
X
X
X
Square
X
X
X
X
X
Exactly one pair of opposite
sides are parallel
X
X
Consecutive sides are
congruent
Consecutive angles are
congruent
supp
Diagonals bisect each other
X
X
X
supp
X
X
X
X
X
X
X
Isosceles
Trapezoid
X
X
Diagonals perpendicular
Diagonals bisect opposite
angles
Trapezoid
X
Activity with cutting out shapes.
Check out the tangram calendars and websites!
Rearrange the 7 pieces back into a SQUARE.
How can the properties of quadrilaterals
be used to solve real life problems?
The tangrams can form 13 convex polygons!
1 triangle
6 quadrilaterals
2 pentagons
4 hexagons
Rearrange the 7 pieces into a TRIANGLE.
Rearrange the 7 pieces into a RECTANGLE.
Rearrange the 7 pieces into a TRAPEZOID.
Rearrange the 7 pieces into a PARALLELOGRAM.
Assignment
Written Exercises on pages 192 & 193
RCOMMENDED: 1 to 9 odd numbers
REQUIRED: 11 to 16 ALL numbers,
26 (NO Proof)
Prepare for Quiz on Lessons 5-4 & 5-5
Tangrams Picture Project
Worth 20 points PLUS possible bonus!
All pictures will be judged to earn
the possible bonus points.
How can the properties of quadrilaterals
be used to solve real life problems?
CLASS Assignment
Written Exercises on page 193
17 to 20 ALL numbers
What is your conclusion?
21 to 25 ALL numbers & 29
What is your conclusion?
How can the properties of quadrilaterals
be used to solve real life problems?
Continue
Prepare for Quiz on Lessons 5-4 & 5-5
Prepare for Test on
Chapter 5: Quadrilaterals
Tangrams Picture Project
Worth 20 points PLUS possible bonus!
All pictures will be judged to earn
the possible bonus points.
How can the properties of quadrilaterals
be used to solve real life problems?