1. Unions, Proofs, and Conditions
A Chapter One Synopsis
By: Giovanni Whitehead

2. Unions and Intersections...
The Union of sets are the elements that make up set one “or” set two, and is represented by
The Intersection of sets are the elements that make up set one “and” set two, and is represented by

3. Example
A good way to understand and differentiate the two is by thinking of a venn diagram.
Given that A={2,4,6,8,10} and B={1,2,3,4,5}
A B={1,2,3,4,5,6,8,10} … the Union
A B={2,4} … the Intersection

4. Unions and Intersections In Geometry
The concept is the same in Geometry, but uses lines, segments, rays, and angles.

5. Two column proofs...
Proofs are a step-by-step explanation that uses theorems, definitions, and postulates to draw a conclusion on a geometric statement.
A theorem is a mathematical statement that can be proven.
A postulate is a statement assumed true without proof; Ex: A line contains at least two points.
In a two column proof, the first column is for statements and the second for reasons (theorems, definitions…)

6. Example o o ~

7. Mentioned Theorems and Definitions
If two angles are (right/straight) angles, then they are congruent
Definitions
A point (or ray, line, and segment) that divides a segment into 2 congruent segments, bisects the segment and the point is called the midpoint of that segment
Two points (or ray, line, and segment) that divides a segment into 3 congruent segments, trisects the segment and the 2 points are called the trisection points of that segment.
A ray (2 rays) that divides an angle into two congruent angles bisects (trisects) the angle, and is called the bisector (trisector) of that angle

8. Paragraph proofs...
Paragraph proofs are often seen in journals, and more advanced mathematics courses. Below is an example of a paragraph proof using the same problem for the two column proof.
<ABC is given to be 90 and <DEF is given to be a right angle, because all right angles measure 90, <DEF is 90 just like <ABC making them = o o o ~

9. Conditional Statements
A conditional statement is a sentence in the form of “If…,Then…”, to make explanations simpler, “If p, then q” will be used (Where p and q are statements).
Negation
To show negation in any statement, the word not (negation of p is not p” is used, to symbolize that, ~ is used.
Converse, Inverse, and Contrapositive
The converse of “If p, then q” would be “If q, then p”
The inverse of “If p, then q” would be “If ~p, then ~q”
The contrapositive of “If p, then q” would be “If ~q, then ~p”
Theorems involved with conditions
If a conditional statement is true, then the contrapositive is also true.

10. Chapter One: Mini Quiz _
Given: <ABC is 90 & EF bisects straight angle DEH
Prove: <ABC = <DEF 1. 2. ~ 3.
Write the converse, inverse, and contrapositive of the following statement: If my hair is yellow, then I am blonde.

11. Theorems, and Definitions
A Chapter Two Synopsis
By: Giovanni Whitehead

12. New Definitions...
Lines, rays, or segments that intersect at right angles are perpendicular (-I-)
Complementary angles are two angles whose sum is 90 (right angle)
Supplementary angles are two angles whose sum is 180 (straight angle)
Two collinear rays that have a common endpoint and extend in different directions are called opposite rays
Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite (relates to a theorem)

13. New Theorems...
If angles are supplementary/complementary to the same angle, then they are congruent.
If angles are supplementary/complementary to congruent angles, they they are congruent.
If a segment/angle is added to two congruent segments/angles, then the sums are congruent (Addition property)
If congruent segments/angles are added to congruent segments/angles, then the sums are congruent (Addition property)

14. New Theorems Cont...
If segments or angles are congruent, then their like multiples or divisions are congruent. (Multiplication/Division Property)
If angles or segments are congruent to the same angle or segment, then they are congruent to each other. (Transitive Property)
If angles or segments are congruent to congruent angles or segments, then they are congruent to each other. (Transitive property)
Vertical angles are congruent.

15. Chapter 2: Mini Quiz Given: QT perpendicular PR m<3 = 45
Prove: m<2 = 40
Given: m<AFB = 60
m<BCD = 30
Prove: <FDC = 90
Given: Line ABCD
m<ECD = 140
m<ABF = 40
Prove: <FBC and <ECB are supplementary.

16. Postulates, and Triangles
A Chapter Three Synopsis
By: Giovanni Whitehead

17. Triangle synopsis...
There are different names for triangles having special characteristics, such as scalene, isosceles, and equilateral.
Scalene - A triangle in which no two side is congruent.
Isosceles - A triangle in which at least two sides are congruent.
Equilateral - A triangle in which all sides are congruent.

18. Congruencies...
In congruent polygons, all pair of corresponding parts are congruent.
Postulate- Any segment or angle is congruent to itself (Reflexive Property).

19. Proving Triangles Congruent...
SSS (Side Side Side) Postulate - If there exists a correspondence between the vertices of two triangles that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
SAS (Side Angle Side) Postulate - If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

20. Triangle postulates Cont...
ASA (Angle Side Angle)Postulate - If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding side of the other, the two triangles are congruent.
HL (Hypotenuse Leg)Postulate - If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding side of the other, the two right triangles are congruent.

21. Angle-Side Theorems...
If two sides of a triangle are congruent, the angles opposite the sides are congruent.
If two angles of a triangle are congruent, the sides opposite the angles are congruent.
This can be applied vice-versa; If two angles/sides of a triangle are not congruent the sides/angles opposite the angles/sides are not congruent

22. CPCTC and Beyond...
CPCTC - is short for “Corresponding Parts of Congruent Triangles are Congruent”
In proofs, when triangles are given to be congruent, it can be useful to have this principle to solve the proof.
Beyond CPCTC
Median - A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the other side. (A median divides/bisects the side to which it is drawn).
Altitude - An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side.(An altitude of a triangle forms right [90] angles with one of the sides.

23. Chapter 3: Mini Quiz

24. Advanced Problem Set, if you dare...
Find X

25. Answers to all problems
Chapter 1
EB , F, AE , BC , BE or BC , <BEA
It is given that <ABC is 90 and EF bisects <DEH, because a bisection divides an angle into two congruent angles, <DEF and <FEH are 90 as well, thus making <DEF congruent <ABC
If I am blonde, my hair is yellow; If my hair is not yellow, I am not blonde; If I am not blonde, my hair is not yellow.
Chapter 2
It is given that m<AFB = 60, and M<BCD = 30,looking at the bigger triangle, it can be said that all angles of a triangle add up to 180 degrees, thus 180 minus the two given angles will give <FDC as 90 degrees
It is given that m<ECD = 140, and m<ABF = 40, because ABCD is a line it is also a straight angle that has to be 180 degrees, thus = <FBC(140) and = <ECB(40), these two angles added together equal 180 degrees, making them supplementary.
It is given that QT is perpen. to PR so it forms right angles measuring 90 degrees. <1 is also given to be 50 degrees, so by angle subtraction ( 90 - <1(50) = <2(40) ) We get that <2 measures 40 degrees.

26. Answer Cont...
Chapter 3, look at the given to know which problem it is.

27. Advanced Problem 1. Calculate some known angles:
ACB = 180-(10+70)-(60+20) = 20°
AEB = (50+30) = 40°
2. Draw a line from point E parallel to AB, labeling the intersection with AC as a new point F and conclude:
FCE ACB
CEF = CBA = = 80°
FEB = = 100°
AEF = = 60°
CFE = CAB = = 80°
EFA = = 100°
3. Draw a line FB labeling the intersection with AE as a new point G and conclude:
AFE BEF
AFB = BEA = 40°
BFE = AEF = 60°
FGE = = 60° = AGB.
ABG = = 60°
4. Draw a line DG. Since AD=AB (leg of isosceles) and AG=AB (leg of equilateral), conclude:
AD = AG.
DAG is isosceles
ADG = AGD = (180-20)/2 = 80°
5. Since DGF = = 40°, conclude:
FDG (with two 40° angles) is isosceles, so DF = DG
6. With EF = EG (legs of equilateral) and DE = DE (same line segment) conclude:
DEF DEG by side-side-side rule
DEF = DEG = x
FEG = 60 = x+x
Answer: x = 30°
1. Calculate some known angles:
ACB = 180-(10+70)-(60+20) = 20°
AEB = (60+20) = 30°
2. Draw a line from point D parallel to AB, labeling the intersection with BC as a new point F and conclude:
DCF ACB
CFD = CBA = = 80°
DFB = = 100°
CDF = CAB = = 80°
ADF = = 100°
BDF = = 60°
3. Draw a line FA labeling the intersection with DB as a new point G and conclude:
ADF BFD
AFD = BDF = 60°
DGF = = 60° = AGB
GAB = = 60°
DFG (with all angles 60°) is equilateral
AGB (with all angles 60°) is equilateral
4. CFA with two 20° angles is isosceles, so FC = FA
5. Draw a line CG, which bisects ACB and conclude:
ACG CAE
FC-CE = FA-AG = FE = FG
FG = FD, so FE = FD
6. With two equal sides, DFE is isosceles and conclude:
DEF = 30+x = (180-80)/2 = 50
Answer: x = 20°