1. 1.1 Statements and Reasoning
Statement – group of words/symbols which is either true or false.
Examples of geometric statements:
mA = 80º
mB + mC = 80º
ABC is a right triangle
Line l is parallel to line m

2. 1.1 Statements and Reasoning
Deduction – the truth of the conclusion is guaranteed. Example:
If p then q p
Therefore q
Induction – the truth is not guaranteed. Example:
3, 5, and 7 are odd numbers that are prime
Therefore all odd numbers are prime
Geometric proofs use deductive logic

3. 1.2 Informal Geometry and Measurement
 Point – represented by a dot
Line – with arrows on each end
Ray – with an arrow on one end
Collinear – 3 points are collinear if they are on the same line. In between – x is in between A and B A x B

4. 1.2 Informal Geometry and MeasurementC 1 B
Angle – may be referred to as ABC, B, or 1 A D
Triangle – referred to as DEF E F
Line segment – referred to as ,
BC = length of the segment B C
Midpoint – if AB = BC, since B is between A and C, B is the midpoint A B C

5. 1.2 Informal Geometry and Measurement
Congruence (denoted by )
Two segments are congruent if they have the same length
Two angles are congruent if they have the same measure
Bisect – to divide into 2 equal parts

6. 1.2 Informal Geometry and Measurement
Bisecting a segment into 2 congruent segments
Bisecting an angle into 2 congruent angles

7. 1.3 Early Definitions and Postulates
Definitions – terminology of the mathematical system is defined. Examples:
Isosceles triangle – a triangle that has 2 congruent sides
Line segment – consists of the 2 points (endpoints) and all the points between them

8. 1.3 Early Definitions and Postulates
Postulates – assumptions necessary to build the mathematical system. Examples:
Postulate 1: Through 2 distinct points, there is exactly one line.
Postulate 2:The measure of any line segment is a unique positive number

9. 1.3 Early Definitions and PostulatesB C
Segment addition – AB + BC = AC
If AC = 8 and BC = 5, what is length AB?

10. 1.4 Angles and their relationships
Angle addition - mABD + mDBC = mABC
If ABC = 130º and mDBC = 50º , what is mABD? A D B C

11. 1.4 Angles and their relationships
Acute angle – 0 < x < 90
Right angle - 90
Obtuse angle – 90 < x < 180
Straight angle - 180

12. 1.4 Geometry Terminology – Pairs of Angles
Complementary angles – add up to 90
Supplementary angles – add up to 180
Vertical angles – the angles opposite each other are congruent

13. 1.5 Introduction to Geometric Proof
Form of a geometric proof:
Statements
Reasons
1. mABC = 80º
1. Given
2. ABC and DBE are vertical angles
2. Given
3. mDBE = 80º
3. Vertical angles have equal measure

14. 1.5 Introduction to Geometric Proof
Examples of Reasons:
Given (use first)
Definitions (like “definition of bisector”)
Properties (like “corresponding angles are congruent”)
Postulates and theorems (like “segment addition”)

15. 1.5 Introduction to Geometric Proof
Properties of equality:
Reflexive (also referred to as “identity”): a = a
Symmetric: if a = b then b = a
Transitive: if a = b and b = c, then a = c

16. 1.5 Introduction to Geometric Proof
Properties of congruence:
Reflexive (also referred to as “identity”): 1  1
Symmetric: if 1  2 then 2  1
Transitive: if 1  2 and 2  3 , then 1  3

17. 2.1 Parallel Lines – Special Angles
Intersection – 2 lines intersect if they have one point in common.
Perpendicular – 2 lines are perpendicular if they intersect and form right angles
Parallel – 2 lines are parallel if they are in the same plane but do not intersect

18. 2.1 Parallel Lines – Special Angles3 4 5 6 7 8
When 2 parallel lines are cut by a transversal the following congruent pairs of angles are formed:
Corresponding angles:1 & 5, 2 & 6, 3 & 7, 4 & 8
Alternate interior angles: 4 & 5, 3 & 6
Alternate exterior angles: 1 & 8, 2 & 7

19. 2.1 Parallel Lines – Special Angles3 4 5 6 7 8
When 2 parallel lines are cut by a transversal the following supplementary pairs of angles are formed:
Same side interior angles: 3 & 5, 4 & 6
Same side exterior angles: 1 & 7, 2 & 8

20. 2.1 Parallel Lines – Special Angles
Terminology:
Corresponding angles – in the same relative “quadrant” (upper right, lower left, etc.)
Alternate – on opposite sides of the transversal
Same side – on the same side of the transversal
Interior – in between the 2 parallel lines
Exterior – outside the 2 parallel lines

21. 2.3 Parallel Lines – Review1 2 3 4 5 6 7 8
What type of angles are:
1 & 8
4 & 6
4 & 5
2 & 6
1 & 7

22. 2.3 Parallel Lines – Review
If 2 lines are parallel and cut by a transversal:
Corresponding angles, alternate interior angles, and alternate exterior angles are congruent
Same-side interior angles and same-side exterior angles are supplementary

23. 2.3 Proving Lines Are Parallel
Given two lines cut by a transversal, if any one of the following are true:
Corresponding angles, alternate interior angles, or alternate exterior angles are congruent
Same-side interior angles or same-side exterior angles are supplementary
Then the two lines are parallel

24. 2.3 Proving Lines Are Parallel: 2 More Theorems
Two lines parallel to the same line must be parallel
Two lines perpendicular to the same line must be parallel

25. 2.4 The Angles of a Triangle
Triangles classified by number of congruent sides
Types of triangles
# sides congruent
scalene
isosceles 2
equilateral 3

26. 2.4 The Angles of a Triangle
Triangles classified by angles
Types of triangles
Angles
acute
All angles acute
obtuse
One obtuse angle
right
One right angle
equiangular
All angles congruent

27. 2.4 Angles of a Triangle
In a triangle, the sum of the interior angle measures is 180º (mA + mB + mC = 180º) A B C

28. 2.4 The Angles of a Triangle
The measure of an exterior angle of a triangle equals the sum of the measures of the 2 non-adjacent interior angles - m1 + m2 = m4 2 1 3 4

29. Question: What do you call a parrot who just died?

30. 2.5 Convex Polygons
Polygon - a closed plane figure whose sides are line segments that intersect only at the endpoints
Regular Polygon – a polygon with all sides equal length and all interior angles equal measure

31. 2.5 Convex Polygons
Concave polygons: A line segment can be drawn between 2 points and the segment is outside the polygon
Convex polygons: A polygon that is not concave

32. 2.5 Convex Polygons Classified by number of sides Polygons # of sides
triangle 3
quadrilateral 4
pentagon 5
hexagon 6
heptagon 7
octagon 8
Polygons
# of sides
nonagon 9
decagon 10
dodecagon 12
15-gon 15
n-gon n

33. 2.5 Convex Polygons Formulas for polygons Sides Diagonals
Sum of the measures of interior angles
Sum of the measures of exterior angles n

34. 2.5 Convex Polygons Formulas for regular polygons Sides
Measure of interior angle
Measure of exterior angle n

35. 3.1 Congruent Triangles
ABC  DEF if all 3 angles are congruent and all 3 sides are congruent.
This means
AB = DE, BC = EF, and AC = DF
ABC   DEF, BAC   EDF and ACB   DFE

36. 3.1 Congruent Triangles
Included/opposite sides and angles for ABC are:
A is opposite side BC
A is included by sides AB and AC
Side AB is opposite C
Side AB is included by A and B C A B

37. 3.1 Congruent Triangles
SSS – If the 3 sides of a triangle are congruent to the 3 sides of a second triangle, then the triangles are congruent
SAS – If 2 sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

38. 3.1 Congruent Triangles
ASA - If 2 angles and the included side of a triangle are congruent to the two angles and included side of a second triangle, then the triangles are congruent.
AAS - If two angles and the non-included side of a triangle are congruent to 2 angles and the non-included side of another triangle, the triangles are congruent

39. 3.1 Congruent Triangles
Right Triangle – In a right triangle, the side opposite the right angle is the hypotenuse and the sides of the right angle are the legs of the right triangle.
HL (hypotenuse-leg) – If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. Note: In the book this is introduced in section 3.2.

40. 3.1 Congruent Triangles To show congruence of triangles: Valid Invalid
SSS
AAA
SAS
SSA
ASA
AAS HL

41. 3.2 Corresponding Parts of Congruent Triangles are Congruent
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
Proofs using CPCTC:
Recognize that what you are trying to prove involves corresponding parts of 2 triangles
Show the triangles are congruent by SSS, SAS, ASA, AAS, etc.
State the conclusion with reason “CPCTC”

42. 3.3 Isosceles Triangles Parts of the isosceles triangle: Vertex
Vertex Angle
Leg
Leg
Base
Base Angles

43. 3.3 Isosceles Triangles
2 sides (legs) of an Isosceles triangle are  (by definition)
2 angles (base angles) of a Isosceles triangle are 

44. 3.3 Equilateral Triangles
An equilateral triangle is also equiangular
An equiangular triangle is also equilateral
Each angle of an equilateral triangle measures 60
60
60
60

45. 3.3 Triangle Terminology
Angle bisector: divides an angle of the triangle into two equal angles
Median: segment that connects a vertex of a triangle to the midpoint of the other side

46. 3.3 Triangle Terminology
Altitude: line segment drawn from the vertex of a triangle that is perpendicular to the opposite side (note: the altitude can be outside the triangle)
Perpendicular bisector: (of a side of a triangle) is the line that intersects the midpoint of the side and is perpendicular to the side

47. 3.4 Three Basic Constructions
Construct the perpendicular bisector (first half of problem 15)
Construct the angle bisector (problem 9 and second half of problem 15)
Construct an angle with the same measure with a given ray/segment as one of the sides (problem 7)
Note: trick to get a 60 degree angle is to construct an equilateral triangle

48. 3.5 Inequalities in a Triangle
The angle opposite the larger side is the bigger angle. In ABC, if AB > AC then m C > m B A B C

49. 3.5 Inequalities in a Triangle
The side opposite the larger angle is the bigger side. In ABC, if m C > m B then AB > AC A B C

50. 3.5 Inequalities in a Triangle
Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side In ABC, CA + AB > BC A B C