Chapter 2 Justification and Similarity 2.1 Students will review triangle congruence conditions and angle relationships. Students will write proofs and learn by contradiction.

Investigation 2.3 Ways to prove triangles congruent Using ruler and or compass to try and construct the given triangles, each person in you group make one and then compare to group Sides 4 in 5 in and 8 in Angles 30, 70, 80 Side AB 5 Side AC 8 and angle A 60 degrees Angle A 50, Angle B 80 side AB 6 Angle A 40, Angle B 70 and side AC 8 Side AB 6 Side AC 9 and angle C 60 degrees When everyone used the same measurements were the triangles congruent What short cuts did you notice to prove triangles congruent

Side Side Side Construct a triangle with sides 2 in, 3 in and 4 in If 3 sides of one triangle are congruent to 3 corresponding sides of another triangle then the triangles are congruent What do you notice about the triangles others created?

Side Angle Side Construct a triangle with an angle of 40 degrees and the two sides of the angle 3 and 4 inches What do you notice about the triangles of those around you? If two sides and the included angle of one triangle are congruent to two corresponding sides and the included angle of another triangle then the two triangles are congruent Included angle – angle between the two given sides

Angle Side Angle What do you think will be true if we construct a triangle knowing 2 angles and the side connecting them? If two angles and the included sides of one triangle are congruent to two angles and the included side of another triangle then the triangles are congruent Included side – side between the two given angles

Side Side Angle What do you think will happened when we construct a triangle with 2 sides and the angle not between them? This one does not work This spells a bad work backwards and we can’t use bad words in this class so it can’t be used

Triangles and Review To name a triangle you need to use the three vertices of the triangle, in any order and put the triangle symbol in front

2.1.1 Triangle Congruence Theorem Corresponding? Same location Congruent? All corresponding parts are congruent Congruence statement? like an equation stating one triangle is congruent to another, order is important, they represent the corresponding and congruent angles

Examples

Investigation 2.1 How did you determine all the angles congruent How did you determine all the sides congruent Do we need all the angles Do we need all the sides What is the minimum amount of information we need to prove triangles congruent Work through 2.2

Homework Worksheet on congruent triangles and congruence statements

2.1.2 Flowcharts for Congruence (Two column proofs) Using flowcharts, paragraph proofs or two column proofs how can you support why triangles are congruent I do more 2 column proofs, I think they are easier but I will show you other methods and then you get to choose which makes the most sense for you, all of them have the same basic concepts Try to do investigations 2.12 and 2.13

2 Column Proof Statement – usually what is congruent, or measurements Reason – why you can write the statement State what is given in the problem, either already marked in the picture or they tell you State other parts of triangles congruent and give a reason why Vertical angles Definitions such as midpoint or bisect Reflexive property – side shared in 2 triangles, could be a common angle Symmetric property a=b and b=a Transitive property a=b, b=c therefore a=c State triangles congruent, congruency statement and why congruent

Two Column Proof Broken into 3 parts Given Make sure you have 3 congruent parts Congruency statement and conjecture

FlowChart and Paragraph proof Flow Chart - Similar to a 2 column proof, more arrows and how things are tied together Paragraph – write out in complete sentences, all the same information needs to be there

Doing a Proof You normally have a picture to go along with the proof. As you state things congruent mark them in the picture As you state things congruent write S or A next to the statement, this helps you keep track of how many parts you have congruent

More on Proofs When writing a proof you will usually need a minimum of 4 statements and reasons. 3 of them are to state sides or angles congruent 1 of them is to state triangles congruent and why

Statements Anything that is told to you in the problem should be written down segment AB bisects segment DC M is the midpoint When adding things together such as angles or a triangle Stating things congruent because of parallel lines

Reasoning Why you are writing the statements Given information Terms we have used on why angles are congruent – vertical angles, AI, AE, Corr if lines are parallel Midpoint proves two segments congruent Angle Bisector, Perpendicular Bisector Reflexive – means it is a side or angle that is used in both triangles Substitution – putting one value in place of another to solve

This is on page 201 Try and write a proof on why the angles add to 180

The proof Statement Reason EC is a line Given AB is parallel to EC 1+2+3=180 def of a line 1=4 AI 3=5 4+2+5=180 Substitution

Examples Given: Prove: Statement Reason AB=CD <ABC=<DCB BC=BC

Statement Reason EC = AC Given ER = AR RC = RC Reflexive REC = RAC SSS Prove the Triangles Congruent Statement Reason EC = AC Given ER = AR RC = RC Reflexive REC = RAC SSS

Example Given: Segment WX and segment YZ bisect each other Prove: the two triangles are congruent

Homework Worksheet on proofs Any method to prove triangles congruent

2.1.3 Converse Idea to know theorem forwards and backwards Writing it different ways depending on what is given and what you need to prove If hypothesis then conclusion Switch the hypothesis and conclusion Converse is not always true Investigation 2.23 and 2.24 For a thereom to be true it always needs to be try, therefore if I can come up with one example to prove it false then it is always false

Systems Review 3 methods to solve systems of equations Can you explain these methods Graphically Substitution Elimination What does it mean to solve a system

Homework Pg 89 2-27 and review 2-28

Probabilty Event: Any outcome or set of outcomes from a situation Successful Event: Set of all outcome sthat are of interest in a given situation Rolling a die Rolling a 5 is an event If you only want and even number rolling a {2,4,6} would be a successful even

Probability Sample Space: all possible outcomes Rolling a die {1,2,3,4,5,6} Probability: likelihood an event will occur, written as fractions or percent Rolling a 5 1/6 only 1 five out of 6 numbers Theoretical Probability: probability that is mathematically calculated Number of successful outcomes/total possible

Homework Pg 202 2-45