Download x = y

Ch. 1
Midpoint of a segment
• The point that divides the segment into two
congruent segments.
3
3
A
P
B
Bisector of a segment
• A line, segment, ray or plane that intersects
the segment at its midpoint.
3
3
A
P
B
Bisector of an Angle
• The ray that divides the angle into two
congruent adjacent angles (pg 19)
More Definitions
• Intersect –
Two or more figures intersect if they
have one or more points in common.
• Intersection –
All points or sets of points the
figures have in common.
When a line and a point intersect,
their intersection is a point.
l
B
When 2 lines intersect, their
intersection is a point.
When 2 planes intersect, their
intersection is a line.
When a line and plane intersect,
their intersection is a point.
Segment Addition Postulate
• If B is between A and C, then AB + BC = AC.
Angle Addition Postulate
• If point B lies in the interior of  AOC,
– then m  AOB + m  BOC = m  AOC.
– What is the interior of an angle?
If  AOC is a straight angle
and B is any point not on AC, then m  AOB + m
 BOC = 180.
Why does it add up to 180?
Ch. 2
The If-Then Statement
Conditional: is a two part statement with an actual
or implied if-then.
If p, then q.
p ---> q
hypothesis conclusion
If the sun is shining, then it is
daytime.
• Circle the hypothesis and underline the
conclusion
If a = b, then a + c = b + c
Other Forms
•
•
•
•
If p, then q
p implies q
p only if q
q if p
Conditional statements are not
always written with the “if”
clause first.
All of these conditionals mean
the same thing.
What do you notice?
Properties of Equality
Numbers, variables, lengths, and angle measures
Addition
Property
if x = y, then x + z = y + z.
Subtraction
Property
if x = y, then x – z = y – z.
Multiplication
Property
if x = y, then xz = yz.
Division
Property
if x = y, and z ≠ 0, then x/z = y/z.
Substitution
Property
Reflexive
Property
Symmetric
Property
Transitive
Property
if x = y, then either x or y may be
substituted for the other in any
equation.
x = x.
A number equals itself.
if x = y, then y = x.
Order of equality does not matter.
if x = y and y = z, then x = z.
Two numbers equal to the same
number are equal to each other.
Properties of Congruence
Segments, angles and polygons
Reflexive
Property
Symmetric
Property
Transitive
Property
AB ≅ AB
A segment (or angle) is congruent to
itself
If AB ≅ CD, then CD ≅ AB
Order of equality does not matter.
If AB ≅ CD and CD ≅ EF, then AB
≅ EF
Two segments (or angles) congruent
to the same segment (or angle) are
congruent to each other.
Complimentary Angles
Any two angles whose measures add up to
90.
If mABC + m SXT = 90, then
 ABC and  SXT are complimentary. S
A
 ABC is the
complement of  SXT
 SXT is the
complement of  ABC
X
B
C
T
See It!
Supplementary Angles
Any two angles whose measures sum to 180.
If mABC + m SXT = 180, then
 ABC and  SXT are supplementary.
S
A
 ABC is the
supplement of  SXT
 SXT is the
supplement of  ABC
X
C
B
T
See It!
Theorem
If two angles are supplementary to congruent
angles (the same angle) then they are
congruent.
If 1 suppl  2 and  2 suppl  3, then
 1   3.
1
2
3
Theorem
If two angles are complimentary to congruent
angles (or to the same angle) then they are
congruent.
If 1 compl  2 and  2 compl  3, then
 1   3.
1
2
3
Theorem
Vertical angles are congruent
(The definition of Vert.
angles does not tell us anything about congruency… this theorem
proves that they are.)
1
4
2
3
Perpendicular Lines ()
Two lines that intersect to form right angles.
If l  m, then
l
angles are right.
m
See It!
Theorem
If two lines are perpendicular, then they form
congruent, adjacent angles.
l
If l  m, then
1   2.
1
2
m
Theorem
If two lines intersect to form congruent,
adjacent angles, then the lines are
perpendicular.
l
If 1   2, then
l  m.
1
2
m
Ch. 3
Parallel Lines (
or
)
The way that we mark that two lines are
parallel is by putting arrows on the lines.
m || n
m
n
Skew Lines ( no symbol  )
Non-coplanar, non-intersecting lines.
q
p
What is the difference between the definition of
parallel and skew lines?
Parallel Planes
Planes that do not intersect.
P
Q
Can a plane and a line be parallel?
Postulate
If two parallel lines are cut by a
transversal, then corresponding angles
t
r
are congruent.
1
2
4
3
Can you name the
corresponding angles?
5
7
6
8
s
Theorem
If two parallel lines are cut by a
transversal, then alternate interior angles
t
are congruent.
r
1
2
4
3
5
7
6
8
s
Theorem
If two parallel lines are cut by a
transversal, then same side interior
t
angles are supplementary.
1
2
4
3
5
7
r
6
8
s
Ways to Prove Lines are Parallel
(pg. 85)
1.
2.
3.
4.
5.
Show that corresponding angles are
congruent
Show that alternate interior angles are
congruent
Show that same side interior angles are
supplementary
In a plane, show that two lines are
perpendicular to the same line
Show that two lines are parallel to a third
line
Types of Triangles
(by sides)
Isosceles
2
congruen
t sides
Equilateral
All sides
congruent
Scalene
No
congruent
sides
Types of Triangles
(by angles)
Equiangular
Acute
Right
Obtuse
3 acute
angels
1 right angle
1 obtuse
angle
Theorem
The sum of the measures of the angles of a
triangle is 180
B
mA + mB + mC = 180
C
A
See It!
Corollary
3. In a triangle, there can be at most one
_right____ or obtuse angle.
Theorem
The measure of an exterior angle of a triangle
equals the sum of the measures of the two
remote interior angles
m1 = m3 + m4
See It!
Regular Polygon
 All angles
congruent
 All side congruent
Theorem (pg 102)
The sum of the measures of the interior
angles of a convex polygon with n sides is
(n-2)180.
Theorem
The sum of the measures of the exterior
angles, one at each vertex, of a convex
polygon is 360.
4
1
1
3
3
2
1 + 2 + 3 = 360
2
1 + 2 + 3 + 4 = 360
REGULAR POLYGONS
• All the interior angles are congruent
• All of the exterior angles are congruent
(n-2)180
n
360
n
= the measure of each
interior angle
= the measure of each
exterior angle
Problems for Ch. 1 - 3
• 1 – 7 - 8 – 10
• 15 – 16 – 17
• 22 – 24 – 25 – 26 – 28
• 31 – 34 – 36
Ch. 4
Definition of Congruency
Two polygons are congruent if corresponding
vertices can be matched up so that:
1. All corresponding sides are congruent
2. All corresponding angles are congruent.
ABC  DEF
The order in which you name the triangles matters !
A
E
C
B
F
D
 ABC   XYZ
• Based off this information with or without a
diagram, we can conclude…
• Letters X and A, which appear first, name
corresponding vertices and that
–  X   A.
• The letters Y and B come next, so
–  Y   B and
–XY  AB
Five Ways to Prove  ’s
All Triangles:
ASA
SSS
SAS
AAS
Right Triangles Only:
HL
Isosceles Triangle
By definition, it is a triangle with two
congruent sides called legs.
X
Legs
Vertex Angle
Base Angles
Y
Z
Base
Theorem
If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
B
A
C
Conclusions
1.  R   S
2. PQ bisects  RPS
P
3. PQ bisects RS
4. PQ  RS at Q
5. PR  PS
S
R
Q
Given: MK OK;
KJ bisects  MKO;
M
Prove: JK bisects  MJO
J
1
2
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK;
KJ bisects  MKO
2.  3   4
1. Given
3. JK  JK
3. Reflexive Property
4.  MKJ  OKJ
4. SAS Postulate
5.  1   2
5. CPCTC
6. JK bisects  MJO
6. Def of  bisector
2. Def of  bisector
K
Ch. 5
Parallelograms: What we now
know…
• From the definition..
1. Both pairs of opposite sides are parallel
• From theorems…
1. Both pairs of opposite sides are congruent
2. Both pairs of opposite angles are congruent
3. The diagonals of a parallelogram bisect each
other
Five ways to prove a Quadrilateral
is a Parallelogram
1. Show that both pairs of opposite sides parallel
2. Show that both pairs of opposite sides congruent
3. Show that one pair of opposite sides are both
congruent and parallel
4. Show that both pairs of opposite angles congruent
5. Show that diagonals that bisect each other
Rectangle
By definition, it is a quadrilateral with four
right angles.
R
S
V
T
Theorem
The diagonals of a rectangle are congruent.
WY  XZ
W
Z
P
X
Y
Rhombus
By definition, it is a quadrilateral with four
congruent sides.
B
C
A
D
Theorem
The diagonals of a rhombus are
perpendicular.
K
X
J
L
M
What does the
definition of
perpendicular lines
tell us?
Theorem
Each diagonal of a rhombus bisects the
opposite angles.
K
X
J
L
M
Square
By definition, it is a quadrilateral with four
right angles and four congruent sides.
B
C
What do you notice
about the definition
compared to the
previous two?
The square is the
most specific type
of quadrilateral.
A
D
Trapezoid
A quadrilateral with exactly one pair of
parallel sides.
Trap. ABCD
B
C
How does this definition
differ from that of a
parallelogram?
A
D
The Median of a Trapezoid
A segment that joins the midpoints of
the legs.
B
C
X
Y
Note: this applies to any trapezoid
A
D
Theorem
The median of a trapezoid is parallel to the
bases and its length is the average of the
bases.
Note: this applies to any trapezoid
B
X
C
Y
How do we find an average of the bases ?
A
A
D
Problems For Ch. 4, 5
•
•
•
•
5
11- 19
20 – 29
30 – 33 – 35 – 37 – 38 – 39 - 40
Ch. 6
Indirect Proof
• Are used when you can’t use a direct proof.
• BUT, people use indirect proofs everyday to
figure out things in their everyday lives.
• 3 steps EVERYTIME (p. 214 purple box)
Step 1
• “Assume temporarily that….” (the conclusion
is false). I know I always tell you not to
ASSume, but here you can. You want to
believe that the opposite of the conclusion is
true (the prove statement).
Step 2
• Using the given information of anything else that you
already know for sure…..(like postulates, theorems,
and definitions), try and show that the temporary
assumption that you made can’t be true. You are
looking for a contradiction* to the GIVEN
information.
• “This contradicts the given information.”
• Use pictures and write in a paragraph.
Step 3
• Point out that the temporary assumption
must be false, and that the conclusion must
then be true.
• “My temporary assumption is false and…” (
the original conclusion must be true). Restate
the original conclusion.
Given:  XYZW; m  X = 80º
Prove:  XYZW is not a rectangle
Assume temporarily that  XYZW is a
rectangle. Then  XYZW have four right
angles because this is the definition of a
rectangle. This contradicts the given
information that m  X = 80º.* My temporary
assumption is false and  XYZW is not a
rectangle.