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 Definitions Intersect – Intersect – Two or more figures intersect if they have one or more points in common. Intersection – Intersection – All points or sets of points the figures have in common.

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

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. 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 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 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 Angles Any two angles whose measures add up to 90. If m  ABC + 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 Angles Any two angles whose measures sum to 180. If m  ABC + m  SXT = 180, then  ABC and  SXT are supplementary. S X T A B C 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  

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  

Theorem Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.)

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 m 1 2 If l  m, then  1   2.

Theorem If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular. l m 1 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. QP 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 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

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

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 m  A + m  B + m  C = 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 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  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 – – – 16 – – 24 – 25 – 26 – – 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.

matters The order in which you name the triangles matters !  ABC   DEF A B C 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 AAS Right 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 S R 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 StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property 4.  MKJ   OKJ 4. SAS Postulate 5.  1   2 5. CPCTC 6. JK bisects  MJO6. Def of  bisector K O J M A A S S S K O J M

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 T V

Theorem The diagonals of a rectangle are congruent. WY  XZ W XY 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 A A B C D X Y How do we find an average of the bases ? Note: this applies to any trapezoid

Problems For Ch. 4, – – 33 – 35 – 37 – 38 –

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.