Ch. 1

Midpoint of a segment

• The point that divides the segment into two congruent segments.

A

B

P

3

3

Bisector of a segment

• A line, segment, ray or plane that intersects the segment at its midpoint.

A

B

P

3

3

Bisector of an Angle

• The ray that divides the angle into two congruent adjacent angles (pg 19)

More DefinitionsMore Definitions• IntersectIntersect – –

Two or more figures intersect if they Two or more figures intersect if they have one or more points in common.have one or more points in common.

• IntersectionIntersection – – All points or sets of points the All points or sets of points the

figures have in common.figures have in common.

When a line and a point intersect, When a line and a point intersect, their intersection is a point.their intersection is a point.

BB

ll

When 2 lines intersect, their When 2 lines intersect, their intersection is a point.intersection is a point.

When 2 planes intersect, their When 2 planes intersect, their intersection is a line.intersection is a line.

When a line and plane intersect, When a line and plane intersect, their intersection is a point.their intersection is a point.

Segment Addition Postulate

• If B is between A and C, then AB + BC = AC. A

C

B

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 angleand 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

What do you notice?

Conditional statements are not always written with the “if” clause first.

All of these conditionals mean the same thing.

Properties of Equality

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.

Numbers, variables, lengths, and angle measures

Substitution Property

if x = y, then either x or y may be substituted for the other in any equation.

Reflexive

Property

x = x.

A number equals itself.

Symmetric Property

if x = y, then y = x.

Order of equality does not matter.

Transitive

Property

if x = y and y = z, then x = z.

Two numbers equal to the same number are equal to each other.

Properties of Congruence

Reflexive

Property

AB AB≅

A segment (or angle) is congruent to itself

Symmetric Property

If AB CD, then CD AB≅ ≅

Order of equality does not matter.

Transitive

Property

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.

Segments, angles and polygons

Complimentary AnglesAny two angles whose measures add up to 90.If mABC + m SXT = 90, then ABC and SXT are complimentary.

S

X

T

A

B C

See It!

ABC is the complement of SXT SXT is the

complement of ABC

Supplementary AnglesAny two angles whose measures sum to 180.If mABC + m SXT = 180, then ABC and SXT are supplementary.

S

X

T

A

BC

See It!

ABC is the supplement of SXT SXT is the

supplement of ABC

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

TheoremVertical angles are congruent (The definition of Vert.

angles does not tell us anything about congruency… this theorem proves that they are.)

1

2

3

4

Perpendicular Lines ()

Two lines that intersect to form right angles.

If l m, then

angles are right.l

m

See It!

Theorem

If two lines are perpendicular, then they form congruent, adjacent angles.

l

m1 2

If l m, then

1 2.

Theorem

If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular.

l

m1 2

If 1 2, then

l 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.

p

q

What is the difference between the definition of parallel and skew lines?

Parallel Planes

Planes that do not intersect.

Q

P

Can a plane and a line be parallel?

Postulate

If two parallel lines are cut by a transversal, then corresponding angles are congruent. r

s

t

1 2

3 4

5 6

7 8

Can you name the corresponding angles?

Theorem

If two parallel lines are cut by a transversal, then alternate interior angles are congruent. r

s

t

1 2

3 4

5 6

7 8

Theorem

If two parallel lines are cut by a transversal, then same side interior angles are supplementary. r

s

t

1 2

3 4

5 6

7 8

Ways to Prove Lines are Parallel (pg. 85)

1. Show that corresponding angles are congruent

2. Show that alternate interior angles are congruent

3. Show that same side interior angles are supplementary

4. In a plane, show that two lines are perpendicular to the same line

5. 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)

Acute

3 acute angels

Right

1 right angle

Obtuse

1 obtuse angle

Equiangular

Theorem

The sum of the measures of the angles of a triangle is 180

A

B

C

mA + mB + mC = 180

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

12

3

4

m1 = m3 + m4

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.

1

23

1

2

3

4

1 + 2 + 3 = 360 1 + 2 + 3 + 4 = 360

REGULAR POLYGONS

• All the interior angles are congruent • All of the exterior angles are congruent

(n-2)180

n= the measure of each interior angle

360

n= 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 congruent2. All corresponding angles are congruent.

                  

                                          

            

The order in which you name the triangles mattersmatters !

ABC DEF

A

BC

F

E

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 AASRight Triangles Only:HL

Isosceles Triangle

By definition, it is a triangle with two congruent sides called legs.

X

Y Z

Base

Base Angles

Legs Vertex Angle

Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

A

B

C

Conclusions

P

Q

SR

1. R S

2. PQ bisects RPS

3. PQ bisects RS

4. PQ RS at Q

5. PR PS

Given: MK OK;KJ bisects MKO;

Prove: JK bisects MJO

Statements Reasons

1. MK OK; KJ bisects MKO

1. Given

2. 3 4 2. Def of bisector

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

K

O

J

M

1

2

34

K

O

J

M

1

2

34

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 parallel2. Show that both pairs of opposite sides congruent3. Show that one pair of opposite sides are both

congruent and parallel4. Show that both pairs of opposite angles congruent5. Show that diagonals that bisect each other

Rectangle

By definition, it is a quadrilateral with four right angles.

R

S T

V

TheoremThe diagonals of a rectangle are congruent.WY XZ

W

X Y

Z

P

Rhombus

By definition, it is a quadrilateral with four congruent sides.

A

B C

D

Theorem

The diagonals of a rhombus are perpendicular.

J

K

L

M

X

What does the definition of perpendicular lines tell us?

Theorem

Each diagonal of a rhombus bisects the opposite angles.

J

K

L

M

X

Square

By definition, it is a quadrilateral with four right angles and four congruent sides.

A

B C

D

The square is the most specific type of quadrilateral.

What do you notice about the definition compared to the previous two?

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

A

B C

D

Trap. ABCD

How does this definition differ from that of a parallelogram?

The Median of a Trapezoid

A segment that joins the midpoints of the legs.

A

B C

D

X Y

Note: this applies to any trapezoid

Theorem

The median of a trapezoid is parallel to the bases and its length is the average of the bases.

B C

D

X Y

AA

B C

D

X Y

How do we find an average of the bases ?

Note: this applies to any trapezoid

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 = 80ºProve: 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.