Understanding Triangle Congruence and Proofs in Geometry

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

2. 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

3. 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

4. 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

5. 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

6. Homework Worksheet on congruent triangles and congruence statements

7. 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

8. 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

9. 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

10. Homework Worksheet on proofs Any method to prove triangles congruent

11. 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

12. 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

13. Homework Pg89 2-27 and review 2-28

14. 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

15. 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

16. Homework Pg 202 2-45