I Feel Alive with Chapter 5
All proofs must be done on a separate piece of paper for credit. As always,
the standard on a review sheet is 100%. You will not receive credit for
mediocrity.
Definitions:
Parallelogram, Rectangle, Rhombus, Square, Trapezoid, midpoint connector, isosceles
trapezoid, median of a Trapezoid
Theorems and definitions:
Opposite sides of a parallelogram are parallel (definition)
Opposite sides of parallelogram congruent
Opposite angles of parallelogram congruent
Diagonals of a parallelogram bisect each other
If both pairs of opposite sides congruent, then parallelogram
If one pair of opposite sides both parallel and congruent, then parallelogram
If both pairs of opposite angles congruent, then parallelogram
If diagonals of quadrilateral bisect each other, then parallelogram
If both pairs of opposite sides parallel, then parallelogram
If 2 lines parallel, equidistant at all points
3 lines parallel, cut off congruent segments on transversals
: Line parallel to one side containing the midpoint of another passes through opposite
midpoint
: Segment through 2 midpoints (midpoint connector): parallel to third side, ½ length
of third side
Diagonals of rectangle congruent
Diagonals of rhombus perpendicular
Diagonals of rhombus bisect opposite angles
Midpoint of hypotenuse of right triangle equidistant from all three vertices
Right angle in parallelogram forms a rectangle
2 Consecutive sides of parallelogram congruent, then rhombus
Trapezoid: Quadrilateral with exactly one pair of parallel lines
Isosceles trapezoid: Both pairs of base angles are congruent, legs congruent
Median of trapezoid: parallel to base, ½(base 1 + base 2) - the average of the bases
T
X
U
M
N
V
W
1. In isosceles trapezoid TUWV, and the median shown MN , MX is parallel to UW .
 T = 60, UW = 4 and the perimeter of MXUN = 46. Find the perimeter of TUWV.
U
N
M
V
P
X
W
2.
Given: UVWX is a parallelogram, M and N are midpoints of UX and UV
Prove: MNVP is a parallelogram.
J
K
L
H
I
3.  HIJ is a right angle, and K and L are the midpoints of the segments as shown.
If KL = x, KI = 2y – 1, JH = 2x + 4, and HI = 3y + 3, find the values of x and y.
M
R
P
S
Q
N
O
4. MN and MO are trisected as shown where MR = RS = SN and
MP = PQ = QO.
If ON = 36, find RP and SQ.
T
X
V
U
W
5. TUWV is an isosceles trapezoid with parallelogram TXWV and  V = 115.
Find  UWX.
6. Answer with always, sometimes or never:
a. A trapezoid has 3 congruent sides ______________
b. A rectangle has diagonals that bisect angles. _____________
c. A rhombus has 4 congruent angles _____________
d. A trapezoid has diagonals that are perpendicular _____________
e. Consecutive angles of trapezoid are supplementary.
7. Prove the statement: If the diagonals of a trapezoid are congruent, then the
trapezoid is isosceles. Figure out the given and prove. (Hint: drop 2
perpendicular lines from A and C, creating 2 right triangles.)
A
C
G
E
Given: ____________________________
Prove: Trapezoid ACEG is an isosceles trapezoid.
A
C
L
G
E
8. Given isosceles trapezoid ACEG.
Prove: ALG  CLE
9.
ABCD is a parallelogram with perimeter 32. AB = 4x + z, BC = 6x + 3z,
CD = 2y + 1 and AD = 3y. Find x, y and z.
T
S
X
1
W
6
P
2
3
5
4
Q
V
10.
Given: QW bisects TQV ; VX bisects SVQ ; STQV is a parallelogram
Prove: WXQV is a rhombus