M 1312
Section 4.4
1
Trapezoids
Definition: A trapezoid is a quadrilateral with exactly two parallel sides.
Parts of a trapezoid:
Leg
Base

Leg
Leg
Base
Base
 Base

Leg
Isosceles Trapezoid:
Every trapezoid contains two pairs of consecutive angles that are supplementary.
Definition:
An altitude of a trapezoid is a segment drawn from any point on one of the parallel sides (base)
perpendicular to the opposite side (the other base).
An infinite number of altitudes may be drawn in a trapezoid.
M 1312
Section 4.4
2
Definition:
A median of a trapezoid is the segment that joins the midpoints of the nonparallel sides (legs).
Median
Median
Theorem: The median of a trapezoid is parallel to each base and the length of the median equals
one-half the sum of the lengths of the two bases.
M
Definition:
An isosceles trapezoid is a trapezoid in which the legs (nonparallel sides) are congruent.
An isosceles trapezoid features some special properties not found in all trapezoids.
Theorem 4.4.1: The base angles of an isosceles trapezoid are congruent.
Theorem 4.4.2: The diagonals of an isosceles trapezoid are congruent.
M 1312
Section 4.4
3
Properties of Isosceles Trapezoid
1. The legs are congruent.
2. The bases are parallel.
3. The lower base angles of an isosceles trapezoid are congruent.
4. The upper base angles of an isosceles trapezoid are congruent.
5. The lower base angle is supplementary to any upper base angle.
6. The diagonals of an isosceles trapezoid are congruent.
7. The median is parallel to the base.
8. The length of the median equals one-half the sum of the lengths of the two bases.
Proving that Trapezoid is isosceles
1. If legs of a trapezoid are congruent then it is an isosceles trapezoid.
2. If two base angles of a trapezoid are congruent, then it is an isosceles trapezoid.
3. If the diagonals of a trapezoid are congruent, then it is an isosceles trapezoid.
Example 1: Given the trapezoid HLJK
H
J
If the mJ  65 and the mK  95 , the measure of angles H and L .
L
K
M 1312
Section 4.4
4
Popper 13 question 1: Given a kite ABCD. AC is the perpendicular bisector of BD. Find BC if
AB = 5 and the perimeter of the kite is 24.
A
D
O
B
C
A. 5 B. 7
C. 10 D 14 E. None of these
Example 2:
Use Isosceles Trapezoid ABCD with length of AD = BC.
D
C
AB ll CD
A
B
a.
mDAB = 75. Find the mADC.
b.
AC
c.
If mA  6 x  25 and mB  8x  15 , find the measures of angle C and D.
= 40. Find BD .
M 1312
Section 4.4
Definition: Am altitude is a line segment from one vertex of one base of the trapezoid and
perpendicular to the opposite base.
5
Popper 13 question 2: In parallelogram ABCD (not shown), the diagonals have the lengths AC = 7
and BD = 9. Which pair of angles have greater measures?
A A and D
B. B and C C. A and C D. A and B E. None of these
Theorem 4.4.3: The length of the median of a trapezoid equals one-half the sum of the bases.
m
1
b1  b2 
2
Example 3:
B
7
I
Find the missing measures of the given trapezoid.
a. mIRD
C
A
b. YR
c. DR
d. AC
D
75
3
X
Y
R
M 1312
Section 4.4
6
Example 4:
HJKL is an isosceles trapezoid with bases HJ and LK , and median RS . Use the given information
to solve each problem.
a.
= 30
= 42
find RS
LK
HJ
L
K
R
b.
= 17
HJ = 14
find LK
c.
=x+5
HJ + LK = 4x + 6
find RS
RS
S
J
H
RS
Example 5:
Given WXYZ is a trapezoid with WX || ZY , MN is the median
W
X
M
Z
N
Y
a. If WX =19 and ZY = 31, find MN
b. If WX = 4x ̶ 7, MN = 2x + 10 and ZY = 2x + 1, find x and the lengths of WX, MN
and ZY.
M 1312
Section 4.4
7
SUMMARY CHARTS:
Special
Quadrilateral
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles Trapezoid
Diagonals Are Always
Congruent
Perpendicular
No
No
Yes
No
No
Yes
Yes
Yes
No
No
Yes
No
Diagonals Always Bisect
Each Other
Angles
Yes
No
Yes
No
Yes
Yes
Yes
Yes
No
No
No
No
There is an excellent chart in your book on page 205.
Use for Popper 13 questions 3-5:
HJKL is an isosceles trapezoid with bases HJ and LK , and median RS . Use the given information
to solve each problem.
L
K
R
S
H
Popper 13 question 3: If LK =30 and HJ = 42 then find RS.
A. RS =36 B RS = 20 C. RS =7 D. RS =2 E. None of these
Popper 13 question 4: If RS =17 and HJ = 14, find LK.
A. RS =36 B RS = 20 C. RS =7 D. RS =2 E. None of these
Popper 13 question 5: If RS = x + 5 and HJ + LK = 4x + 6 find RS.
A. RS =36 B RS = 20 C. RS =7 D. RS =2 E. None of these
J
M 1312
Section 4.4
8
Solutions to Poppers 11and 12:
Use for Popper 11 questions 1 and 2: Assuming ABCD is a parallelogram, find x and y.
A
2y-2
B
2x
D
perimeter is 40
3x
C
Popper 11 question 1. Find the value of x:
A. 7 B. 8 C. 4 D. 12 E None of These
Popper 11 question 2: Find the value of y.
A. 8 B. 7 C. Need more information D. None of theses
Popper 11 question 3: Assume that X, Y, and Z are midpoints of the sides of ∆RST. If RS = 28, ST = 12, and
RT = 18, find XY.
A. XY=6
B. XY = 9 C. XY = 14 D. XY = 24 E. None of these
M 1312
Section 4.4
9
Use for Popper 11` questions 4 and 5:
Given : AB =18 , AD = 9 , AE = 8 and BC = 18 and D and E are midpoints.
C
E
A
D
B
Popper 11 question 4: Find the length of DE.
A. DE = 8 B. DE = 15 C. DE = 18 D. DE = 16 E None of these
Popper 11 question 5: Find the length of EC.
A. EC = 8 B. EC = 15 C. EC = 18 D. EC = 16 E None of these
Use for Popper 12 questions 1 and 2. If point D is midpoint of AB and E is the midpoint of AC
A
9
18
D
8
y
E
x
B
Popper 12 question 1: Find the value for x.
A. 15 B. 9 C. 8 D. 16 E None of these
Popper 12 question 2: Find the value for y.
A. 15 B. 9 C. 8 D. 16 E None of these
30
C
M 1312
Section 4.4
10
Popper 12 question 3: ABCD is a parallelogram. Given that AB = 11x + 6, BC = 12x + 7, and
CD = 13x + 2, find the length of AD.
A. 28
B. 67
C. 31 D. 62 E None of these
Popper 12 question 4: ABCD is a parallelogram. Given that m
find the measure of angle C.
A,
B.
C.
D.
Popper 12 question 5: Given a kite
A = 2x + 8 and m
E. None of These
A
ABCD.
O
D
B
C
AC is the perpendicular bisector of BD. Find AD if AO =4 and BD = 6.
A. AD = 4
B. AD = 6
C. AD = 5 D. AD =25
E. None of these
B = 3x − 28,