Download Chapter 5

5.1 notes
Sunday, November 02, 2008
3:09 PM
Indirect proof procedure.
1. List the possibilities for the conclusion.
2. Assume the negation of the desired conclusion is correct.
3. Write a chain of reasons until you reach an impossibility.
This will be a contradiction of either
a. Given information
b. A theorem, definition, or other known fact.
4. State the remaining possibility as the desired conclusion.
Example 1:
Given : A  D , AB  DE , AC  DF
Pr ove : B  E
D
A
F
B
C
Proof: Either B  E or B  E.
Assume B  E
From the given inf ormation
A  D, AB  DE. thus, ABC  DEF by ASA.
 AC  DF by CPCTC .
But this is impossible, sin ce AC  DF is given.
Thus our assumption is false and B  E , because
this is the only other possibility.
Example 2:
Given : O, AB  BC
Pr ove : AOB  COB
A
B
O
Notes Page 1
C
E
Assume AOB  COB. AO  CO because All radii are  .
By the reflexive property , BO  BO. Thus , AOB  COB by SAS ,
which means that AB  BC by CPCTC . This is impos si ble because
it contradicts the given fact that AB  BC . Our assumption is false,
AOB  COB because that is the only other possibility .
Notes Page 2
5.2 Notes
Monday, November 03, 2008
11:39 AM
Theorem 30: The measure of an exterior angle of a triangle is
greater than the measure of either remote interior angle.
Theorem 31: If two lines are cut by a transversal such that two
alternate interior angles are congruent, the lines are parallel.
Short hand(Alt. int. 's  || lines.)
1
2
4
m
3
6
5
n
Theorem 32: If two lines are cut by a transversal such that two
alternate exterior angles are congruent, the lines are parallel.
Short hand(Alt. ext. 's  || lines.)
1
2
4
m
3
6
5
n
Theorem 33: If two lines are cut by a transversal such that two
corresponding angles are congruent, the lines are parallel.
Short hand(corr. 's  || lines.)
1
2
4
m
3
6
5
n
Theorem 34: If two angles are cut by a transversal such that
two same side interior angles are supplementary, the lines are
parallel.
Short hand(s-s int. 's are supp. || lines.)
1
2
4
m
3
6
5
n
Notes Page 3
Theorem 35: If two lines are cut by a transversal such that
same side exterior angles are supplementary, the lines are
parallel.
Short hand(s-s ext. 's are supp. || lines.)
1
2
4
m
3
6
n
5
Theorem 36: If two coplanar lines are perpendicular to a third
line, they are parallel.
n
m
t
Example 1: Flow proof
Given: 1 is supplementary to 3
2  3
Pr ove : TP || RA
Y
3
R
A
1
2
T
P
Definition: A parallelogram is a four sided figure with both
pairs of opposite sides parallel.
Example 2: Given: 1  2, PQR  RSP
Prove: PQRS is a parallelogram.
Notes Page 4
Q
R
1
P
3
4
2
S
Statements
Reasons
1. 1  2, PQR  RSP 1. Given
2. PQ || RS
2. Alt. int. 's  || lines
3. 3 4
3. Subtraction prop.
4. QR || PS
4. Alt. int. 's  || lines
5. PQRS is a
5. By definition of parallelogram
Notes Page 5
5.3 Notes
Tuesday, November 04, 2008
9:39 AM
Postulate: Through a point not on a line there is exactly one
parallel to the given line.
A
m
n
Theorem 37: If two parallel lines are cut by a transversal, each
pair of alternate interior angles are congruent.
Short hand(|| Lines alt. int. 's  )
1
2
4
m
3
6
n
5
Theorem 38: If two parallel lines are cut by a transversal, then
any pair of the angles formed are either congruent or
supplementary.
1
2
4
m
3
6
5
n
Theorem 39: If two parallel lines are cut by a transversal, each
Notes Page 6
Theorem 39: If two parallel lines are cut by a transversal, each
pair of alternate exterior angles are congruent.
Short hand(|| Lines Alt. ext. 's  )
1
2
4
m
3
6
5
n
Theorem 40: If two parallel lines are cut by a transversal, each
pair of corresponding angles are congruent.
Short hand(|| Lines corr. 's  )
1
2
4
m
3
6
5
n
Theorem 41: If two parallel lines are cut by a transversal, each
pair of same-side interior angles are supplementary.
Short hand(|| Lines s-s int. 's are supp.)
1
2
4
m
3
6
5
n
Theorem 42: If two parallel lines are cut by a transversal, each
pair of same-side exterior angles are supplementary.
Short hand(|| Lines s-s ext. 's are supp.)
1
2
m
Notes Page 7
1
2
4
m
3
6
n
5
Theorem 43: In a plane, if a line is perpendicular to one of two
parallel lines, it is perpendicular to the other.
t
m
n
Theorem 44. If two lines are parallel to a third line, they are
parallel to each other.
Short hand: transitive property of || lines
m
third line
n
Example 1: A crook problem. Solve for x.
100 
x
40
Example 2:
Notes Page 8
Example 2:
If c||d, find the m 1
c
1
(3x + 5) 
(2x + 10) 
d
Example 3:
Given : AD || BC
AB  DC
A
D
AB || DC
Pr ove : B  C
C
B
Statements
1. AD || BC , AB  DC , AB || DC
Reasons
1. Given
2. Draw a line through point D
parallel to AB
2. Parallel post.
3. DAE  BEA
3. || Lines alt. int. 's 
4. BAE  DEA
4. || Lines alt. int. 's 
5. AE  AE
5. Reflexive prop.
6. AEB  EAD
6 ASA(3, 5, 4)
7. AB  DE
7. CPCTC
Notes Page 9
8. DE  DC
8. Transitive prop.
9. DEC  C
9.
10. B  DEC
10. || Lines corr. 's 
11. B  C
Transitive prop.
Notes Page 10
5.4 notes
Wednesday, November 05, 2008
9:57 AM
Definitions:
1. A convex polygon is a polygon in which each interior angle has
a measure less than 180.
A
B
C
E
D
2. A non-convex polygon is a polygon in which at least one
interior angle is greater than 180.
A
B
C
E
D
3. A diagonal of a polygon is any segment that connects two
nonconsecutive (nonadjacent) vertices of the polygon.
4. A quadrilateral is a four sided polygon.
A
D
B
C
• The following are special quadrilaterals.
Notes Page 11
5. A parallelogram is a quadrilateral in which both pairs of
opposite sides are parallel.
A
D
B
C
6. A rectangle is a parallelogram in which at least one angle is a
right angle.
A
D
B
C
7. A rhombus is a parallelogram in which at least two consecutive
sides are congruent.
A
D
B
C
8. A kite is a quadrilateral in which two disjoint pairs of
consecutive sides are congruent.
A
D
C
B
9. A square is a parallelogram that is both a rectangle and a
rhombus
A
D
Notes Page 12
A
D
B
C
10. A trapezoid is a quadrilateral with exactly one pair of parallel
sides. The parallel sides are called the bases of the trapezoid.
C
D
A
B
11. An isosceles trapezoid is a trapezoid in which the nonparallel
sides(legs) are congruent. In the figure, A and B are called
the lower base angles and C and D are called the upper
base angles.
C
B
D
A
Notes Page 13
5.5 notes
Thursday, November 06, 2008
11:46 AM
Definition: A parallelogram is a quadrilateral in which both
pairs of opposite sides are parallel.
A
D
B
1.
2.
3.
4.
C
Properties of a parallelogram
The opposite sides are congruent
The opposite angles are congruent
The diagonals bisect each other
Any pair of consecutive angles are supplementary
Definition: A rectangle is a parallelogram in which at least one
angle is a right angle.
A
D
B
C
Properties of a rectangle
1. All properties of a parallelogram apply by definition.
2. All angles are right angles
3. The diagonals are congruent
Definition: A kite is a quadrilateral in which two disjoint pairs
of consecutive sides are congruent.
A
D
C
B
Properties of a Kite
1. The diagonals are perpendicular.
2. One of the diagonals are the perpendicular bisector of the
other.
3. One of the diagonals bisect a pair of opposite angles
4. One pair of opposite angles are congruent.
Definition: A rhombus is a parallelogram in which at least two
consecutive sides are congruent.
A
B
D
C
Notes Page 14
consecutive sides are congruent.
A
B
1.
2.
3.
4.
5.
6.
D
C
Properties of a rhombus
All the properties of a parallelogram apply by definition
All the properties of a kite apply
All the sides are congruent
The diagonals bisect the angle.
The diagonals are perpendicular bisectors of each other.
The diagonals divide the rhombus into four congruent right
triangles.
Definition: A square is a parallelogram that is both a rectangle
and a rhombus
A
B
D
C
Properties of a square
1. All the properties of a rectangle apply by definition.
2. All the properties of a rhombus apply by definition.
3. The diagonals form four 45-45-90 triangles.
4. Definition: A trapezoid is a quadrilateral with exactly one pair
of parallel sides. The parallel sides are called the bases of the
trapezoid.
C
D
A
B
5. Definition: An isosceles trapezoid is a trapezoid in which the
nonparallel sides(legs) are congruent. In the figure, A and B
are called the lower base angles and C and D are called the
upper base angles.
C
B
D
A
Properties of an isosceles trapezoid
1. The lower base angles are congruent
2. The upper base angles are congruent
3. The diagonals are congruent
Notes Page 15
3. The diagonals are congruent
4. Any lower base angle is supplementary to any upper base
angle.
Example 1:
Prove property four of parallelograms. The diagonals of a
parallelogram bisect each other.
Given: ABCD is a
Prove: AC and BD bisect each other.
A
D
2
3
E
1
B
Statements
1. ABCD is a
2. AD || BC
3. 1  2
3  4
4
C
Reasons
1. Given
2. By def. of a
|| lines implies alt. int.
angles are 
4. BC  AD
4. Prop. of a
5. BEC  DEA
5. ASA(3, 4, 3)
6. AE EC
6. CPCTC
BE ED
7. Diagonals bisect
each other
7 by def of segment
bisector
Notes Page 16
5.6 notes
Tuesday, November 11, 2008
8:36 PM
• Methods to prove that quadrilateral ABCD is a parallelogram.
1. If both pairs of opposite sides of a quadrilateral are parallel,
then the quadrilateral is a parallelogram.
Short hand: By definition of a parallelogram.
A
D
B
C
2. If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Short hand: opposite sides are .
A
D
B
C
3. If one pair of opposite sides of a quadrilateral are both parallel
and congruent, then the quadrilateral is a parallelogram.
Short hand: one pair of opposite sides are both || and 
A
D
B
C
Notes Page 17
A
D
B
C
4. If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
Short hand: Diagonals bisect each other.
A
D
B
C
5. If both pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Short hand: opposite are .
A
D
B
C
Example 1:
Given: NRTW is a
NX  TS
WV  PR
Prove: XPSV is a
Notes Page 18
W
V
T
X
S
N
P
Statements
1. NRTW is a
R
Reasons
1. Given
NX  TS
WV  PR
2. N  T, W  R 2. Prop. of a
NR  WT , NW  RT
3.
3. Subtraction prop.
NP  VT
WX  SR
4. NXP  TSV
4. SAS(1, 2, 3)
WXV  RSP
5.
XV  PS
XP  VS
6. XPSV is a
5. CPCTC
6. Opposite sides are 
Notes Page 19
5.7 notes
Wednesday, November 12, 2008
11:31 AM
• Proving that a quadrilateral is a rectangle.
1. If a parallelogram contains at least one right angle, then it is a
rectangle.
Short Hand: By definition of a rectangle.
B
C
A
D
2. If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
Short hand: Diagonals of a parallelogram are congruent.
B
C
A
D
3. If all four angles of a quadrilateral are right angles, then it is a
rectangle.
Short Hand: All four angles are right angles.
B
C
A
D
• Proving that a quadrilateral is a kite
1. If two disjoint pairs of consecutive sides of a quadrilateral are
Notes Page 20
1. If two disjoint pairs of consecutive sides of a quadrilateral are
congruent, then it is a kite.
Short Hand: by definition of kite
B
A
C
D
2. If one of the diagonals of a quadrilateral is the perpendicular
bisector of the other diagonal, then it is a kite.
Short hand: One diagonal is the perpendicular bisector of the
other
B
A
C
D
• Proving that a quadrilateral is a rhombus
1. If a parallelogram contains a pair of consecutive sides that are
congruent, then it is a rhombus.
Short hand: by definition of a Rhombus
B
A
C
D
2. If either diagonal of a parallelogram bisects two angles of the
parallelogram, then it is a rhombus.
Short hand: one diagonal of a parallelogram bisects two angles.
Notes Page 21
B
A
C
D
3. If the diagonals of a quadrilateral are perpendicular bisectors of
each other, then the quadrilateral is a rhombus.
Short hand: diagonals are perpendicular bisectors of each
other.
B
A
C
D
• Proving a quadrilateral is a square.
1. If a quadrilateral is both a rectangle and a rhombus, then it is a
square.
Short hand: by definition of a square
A
D
B
C
• Proving that a trapezoid is isosceles.
1. If the nonparallel sides of a trapezoid are congruent, then it is
isosceles.
Short hand: by definition of an isosceles trapezoid.
A
D
Notes Page 22
A
D
C
B
2. If the lower or upper base angles of a trapezoid are congruent,
then it is isosceles.
Short hand: base angles of a trapezoid are congruent.
A
D
C
B
3. If the diagonals of a trapezoid are congruent, then it is
isosceles.
Short hand: Diagonals of a trapezoid are congruent.
A
D
C
B
Example 1:
What is the most descriptive name for quadrilateral ABCD with
vertices A=(-3, -7), B=(-9, 1), C=(3, 9), and D=(9, 1)?
C(3, 9)
8
6
4
2
Notes Page 23
C(3, 9)
8
6
4
2
D(9, 1)
B(-9, 1)
-5
5
10
15
-2
-4
-6
A(-3, -7)
Slope of AB = -4/3
Slope of BC = 2/3
Slope of CD = -4/3
Slope of AD = 2/3
1. Opposite sides are parallel so it is at least a parallelogram
2. Is it a rectangle?
Since the slopes of AB and BC are not opposite reciprocals of
each other,  ABC is not a right angle. ABCD is not a rectangle.
3. Is it a rhombus?
Check the slopes of the diagonals.
Slope of AC = 8/3
Slope of BD = 0
The slopes are not opposite reciprocals, so the diagonals are
not perpendicular. ABCD is not a rhombus.
ABCD is a parallelogram. :)
Notes Page 24