M 1312
Section 4.3
1
Rectangle
Definition: A rectangle is a parallelogram that has a right angle.
Corollary 4.3.1: All angles of a rectangle are right angles.
Theorem 4.3.2: The diagonals of a rectangle are congruent.
A rectangle is a parallelogram:
1)Oppositesidesarecongruent(theyequaleachother).
2) Oppositeanglesarecongruent(theyequaleachother).
3) Consecutiveanglesaresupplementary(theyaddupto180)
4) Diagonalsbisecteachother(theycuteachotherinhalf)
5) Diagonalsarecongruent(theyequaleachother)
6) Allfouranglesare90.
Example 1:
Given: Rectangle AGPZ
A
G
T
P
a.
GT
= 6. Find AZ .
b.
mAGP = 34. Find mGZA.
c.
AT = 3x + 13 and TG = 5x - 21. Find GP .
Z
M 1312
Example 2: Given the rectangle
Section 4.3
2
M
N
Q
P
a. If QP = 9 and NP = 6, find NQ and MP.
b. If MQ = x , MP = 51 and QP = 2 x, find x and the length of QP.
Example 3:
Given : Rectangle WXYZ with diagonals WY and XZ .
Prove: m1  m2
W
X
V
1
2
Z
Statements
1.
2.
3.
4.
5.
6.
Y
Reasons
Rectangle WXYZ with diagonals WY and XZ
WZ  XY
ZY  ZY
XZY  WYZ
1. Given
2 The diagonals of a rectangle are 
3. Opposite sides of a rectangle are 
4.
5.
6.
M 1312
Section 4.3
3
Example 4: Given: rectangle QRST and Parallelogram QZRC, find length of RZ, ZQ and CS if
RZ = 6x, ZQ = 3x +2y and CS = 14-x
Z
R
Q
TS Example 5: Find the measure of LN. Given LI = 3x-2 and MI = 2x +3 and LMNP is a rectangle.
M
N
I
L
Square
Definition: A square is a rectangle that has two adjacent sides congruent.
Corollary 4.3.3: All sides of a square are congruent.
P
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Section 4.3
4
Example 6:
AGZP is a square with GT = 12. Find AZ .
A
G
T
P
Z
Rhombus
Definition: A rhombus is a parallelogram with two congruent adjacent sides.
Corollary 4.3.4: All sides of a rhombus are congruent.
Theorem 4.3.5: The diagonals of a rhombus are perpendicular.
SquaresandRhombi
Asquareisaquadrilateralwith4rightanglesand4congruentsides.
Arhombusisalsoaquadrilateral,butitscharacterizedby4congruentsides;arhombus
doesNOThavefourcongruentangles.
Thepropertiesofaparallelogramapplytobothsquaresandrhombi.Arhombushowever
hastwospecialproperties:
2) Thediagonalsofarhombusareperpendicular(theyformrightangles)
3) Eachdiagonalofarhombusbisectsapairofoppositeangles(theanglesarecutin
half).
M 1312
Section 4.3
5
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
E
y
x
B
C
30
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
Example 7: Given a rhombus ABCD
A
D
B
a. If DC = 6.3 , find the perimeter of ABCD.
C
b. If DB = 8 and AC = 6, find DC.
M 1312
Section 4.3
6
Example 8:
D
C
ABCD is a rhombus. mADB = 27. Find the mADC.
A
B
Example 9:
FISH is a rhombus with FI = 6x + 2 and SI = 8x - 4. Find FH .
F
H
I
S
Popper12question3:ABCDisaparallelogram.GiventhatAB= 11x+6,BC= 12x+7,and
CD= 13x+2,findthelengthofAD.
A. 28
B. 67
C. 31
D. 62
E None of these
M 1312
Section 4.3
7
Example 10:
Use rhombus ABCD and the given information to find each value.
B
a.
AE = 14
find AC
A
b.
E
C
mABE = 34
find mABC
D
c.
find mDEA
d.
= 4x  1
AB = 20 + x
find “x”
CB
Popper 12 question 4: ABCD is a parallelogram. Given that m ∠ A = 2x + 8 and m ∠ B = 3x − 28,
find the measure of angle C.
A, 92°
B. 88°
C. 80°
D. 52°
E. None of These
M 1312
Section 4.3
Popper 12 question 5: Given a kite
8
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