Name: Geometry Period Unit 5: Congruency Part 1 of 3: Intro to Congruency & Proof Pieces Lessons 5-1 through 5-4 In this unit you must bring the following materials with you to class every day: Please note: Calculator Pencil This Booklet A device You may have random material checks in class Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class every day. All homework for part one of this unit is in this booklet. Answer keys will be posted as usual for each daily lesson on our website
5-1 Notes Today s Goal: How do we explain our thinking in math? Warm-Up Compare the following: Now reflect: a) Which student explained the best? b) What did you like most about the best response? c) What did you notice in the worse response?
Complete the sentence: When you are asked to explain your math, you should What are other actions in math? Which of these would require a full answer in words? Circle them! Let s try one together! 1. In the diagram, it is given thatm 2 m 3. Explain why ABC is an isosceles triangle 1. Because this was 2. Because in a triangle when 3. ABC is an isosceles triangle because
How to Explain in Math 3 Use a theorem or fact to explain why your answer is true Describe an answer or results in detail using accurate and precise math language/notation Describe the different steps taken in the problem solving process 2 recognizes or applies appropriate math symbols and notation Work is produced in an organized structure Clearly labeling or stating a correct answer and key components of a problem Correctly draws a conclusion without reason Explains a response without the use of specific theorems and definitions 1 Doesn t generate a complete or thorough response Incorrectly states a theorem uncovering a misconception 2. Given the following diagram and that E is the midpoint ofbc. a) State a pair of line segments that are congruent. Explain the reason why these two segments are congruent b) State a pair of angles that are congruent. Explain the reason why these two angles are congruent c) If AB BEC what can we conclude about ABE? Explain. Therefore, what type of triangle is ABE?
3. Fact: JH is a perpendicular bisector of KI. a) MARK YOUR DIAGRAM ACCORDINGLY. b) What are 2 facts you know from the statement above? Why are they true? I Know Because 4. Given fact: WY bisects ZWX Mark your diagram. What is ONE relationship we know based on the fact above? Justify your answer I Know Because 5. Based on the markings for each triangle, what is the relationship between ABC and DEF? Justify your answer
6. Consider the diagram of ABCD below. Using proper labels, o o State one pair of angles that are congruent State two pairs of sides that are congruent and 7. Fill in the blanks: (Hint: sometimes you have to read because before answering the question)! Given: m n We Know <2 Because Corresponding Angles are congruent when parallel lines are cut by a transversal <2 + <3 = 180 interior angles are supplementary when parallel lines are cut by a transversal <4 Vertical angles are You choose!! Alternate Interior Angles are congruent when parallel lines are cut by a transversal 8. Fact: ABC DEF Math-Hoo knows: AC DF, because when triangles are congruent, their corresponding sides are the same length You re turn! State a fact you know, why is it true?
5-2 Notes 5-2 Introduction to Proofs and Proof Pieces Learning Goals: What does it mean to Prove and what are Proof tools? Today we will continue our discussion of explaining in Geometry, but we will look more closely at geometric proofs. Proofs are an organized way to display our knowledge! Before we begin our work with proofs we need the right tools to get us there. You will work with partners today. Together, you will use a skill you use in everyday life, but today we will use that skill to make connections to geometry. This is the skill of deductive reasoning. Deductive reasoning uses,, and, to form a logical argument. When you use deductive reasoning, you start with true or given statements and or infer the truth of another statement. Real life example: Your teacher has extra help every Monday after school. Today is Monday. So, what can we deduce/conclude? Geometry Example: Given: B is a midpoint. Based on our mathematical knowledge, what can we mark on our diagram? What can we deduct/conclude? First I can mark Then I know: In your groups, you will be using deduction to mark up diagrams and write some conclusions based on true givens. We will use this toolbox for the rest of the year, so keep it in a safe place!!
Together! Given: (fill in). Given:. Mark Your diagram Now, I know Mark Your diagram Now, I know The proof tools I will use/that are related The proof tool I will use/that are related Given:. Mark Your diagram Now, I know The proof pieces I will use/that are related Do we know anything else, now? The proof piece I will use is
Practice Directions: Using your Proofs tool box- Decide which proof tools would be appropriate for each of the givens and mark your diagram! **You do not have to fill out EVERY box in the charts! (1) Given: WU is a perpendicular bisector. Mark Your diagram Now, I know The proof pieces I will use/that are related (2) Given: ΔZWX is an isosceles triangle, with base ZX and WY bisects ZWX Mark Your diagram Now, I know The proof pieces I will use/that are related
3) Given: HJ is a perpendicular bisector of KI. Mark Your diagram Now, I know The proof pieces I will use/that are related (4) Given: E is the midpoint of BC. Mark Your diagram Now, I know The proof pieces I will use/that are related
5-2 Homework Directions: Complete each of the following problems. Make sure you correct your answers in a different color when you are done! You will need your proof tool box for this homework! For numbers 1-3, a) Read through the givens and make any appropriate markings on the diagram. b) List all the proof tools you might use based on the givens. You may use the name of theorem or the definition. 1. Given: PQR with median QS Mark Your diagram Now, I know The proof pieces I will use/that are related 2. Given: Triangle ABC with BD the altitude to base AC ; D is the midpoint of AC. Mark Your diagram Now, I know The proof pieces I will use/that are related
3. Given: BD is the perpendicular bisector of side AC Mark Your diagram Now, I know The proof pieces I will use/that are related 4. Error Analysis! The question displays the work turned in by a student in Geometry class. John s geometry teacher has notified him that he either forgot to list a proof tool OR he has made a mistake in the markings (marked up something incorrectly OR left out a marking), or both. John needs your help correcting his work! In the space provided- Write down what/ where their mistake is and how to correct it! Start here Given: CS is perpendicular to AT. Correction:
5-3 Notes Today s Goal: What are special features in a proof? How can we practice introductory proofs? Consider the following two proofs from Student A and Student B. After you read them, answer the follow up questions. When you get to the stop sign, stop and wait for a class discussion. Be prepared to share out! Follow up questions: 1. Who do you think received a better grade? Why? Focusing on the BETTER proof only now a) In the better proof, what was the writer trying to prove? Where is this represented in their proof? b) What is the structure of a good proof? c) What details should you show when writing a really good proof?
Before we begin: Let's make a list---- You're about to try some introductory proofs on your own. When you look at a problem, what is the first thing you are going to do? To start a proof One together! Given: ΔPQR with median QS Prove: PS SR
Practice it! Look at the board for tips for ultimate success if you get stuck! Key is available to check your work! 1. Given: AC bisects BAD Prove: BAC DAC Statements Reasons 1. 1. 2. 2. 2. Given: RP bisects AB Prove: MA MB Statements Reasons 1. 1. 2. 2. 3. Given: Lines AB and CD intersect at Point E Prove: AED BEC 1. 1. 2. 2.
4. Given: Triangle PQR with median QS Q Prove: PS SR P S R Statements Reasons 5. Given: Triangle KLM with altitude LP Prove: LPK LPM L K P M Statements Reasons Hey Geometry Friend, Did you check the key yet?! Check now!
6. Given: RSA TSA Prove: AS is an angle bisector. Statements Reasons 7. Given: AD is an altitude Prove: ABD is a right triangle *Longer* 8. Given AB BC AC Prove ABC is an equilateral triangle
9. Given: 2 intersecting segments Prove 1 2 10. Given 1 and 2 are complementary Prove: <A is a right angle A 11. THINK ABOUT IT! Given: RSW TUV Prove: RW TV
5-3 Homework Complete each of the following problems and check the key online! 1. Given: Prove: AB BC Statements Reasons 2. Joseph read the givens in a proof and marked the diagram. Examine his marked diagram below: Which of the following could be one of the Given Statements based on his markings? 1) BD is a median. 2) BD is an altitude. 3) ABC is an isosceles triangle. 4) BD is an angle bisector. WHY? Fully Explain:
3. Given: BD bisects AC Prove: AD CD Statements Reasons 4. Do we know if the following triangles are congruent? Why? 5. Dana is completing the following proof. She did pretty well, however there is something she can do better. Give Dana one good piece of advice to make a better, more accurate and more detailed proof. Given: BD is the altitude in triangle ABC Prove Triangle ADB is a right triangle. Statements 1. BD is the altitude in triangle ABC Reasons 1. Given 2. <ADB is a right angle 2. Altitudes form 90 degree 3. Triangle ADB is a right triangle angles 3. It has a right angle Advice
5-4 Notes 5-4: Triangle Congruence and Proof Pieces Learning Goals: What does it mean for 2 triangles to be congruent? What can we infer based on knowing that 2 triangles are congruent? What important aspects are included in two-column proofs? Warm-Up 1. From our homework last lesson What are some things you can conclude from the diagrams given? *Can we conclude that these two triangles are congruent? Why or why not? *So, if given ΔABC ΔDEF, we can identify the pairs of congruent corresponding parts. Corresponding Angles <A Corresponding Sides AB Therefore, ΔABC is congruent to ΔDEF because <B BC Congruent Triangles are triangles with corresponding angles and sides. Summary of Corresponding Parts
Example 1) If given the following picture, what congruence statement (conclusion) can be made? Justify your answer. because all corresponding and all angles are congruent. Example 2) Mark the following diagram based on the given information. Given: FE//AB, FE AB, FC, CB EC AC Prove: ΔABC ΔEFC Statements 1. FE//AB, FE AB, FC, CB EC AC Reasons 1. Wow! This is really long, maybe one day we will have shortcuts
Continuing Congruency Example 3) Given: ΔABC ΔXYZ Prove AC XZ (Hint: what property to do we need to show congruence?) Statements Reasons PRACTICE! 4. Given: ΔABC ΔXYZ Prove <ACB <XZY. Statements Reasons 5) For parts a-d ΔXYZ ΔMNL. Complete each statement and explain how you got each answer! a. m<y = b. m<m = c. m<z = d. XY =
6) CD bisects AB at P. Prove AP = PB. 7) Given: E is the midpoint of AB. Prove: AE = EB 8) Find m<1: e. b.
9) Consider the following marking. What was the given? 10) BD is a perpendicular bisector of AC. Which statement can NOT always be proven?
11) Given: Isosceles triangle ABC with altitude AB. Prove: ABD CBD. Statement 1. 1. Given Reason 2. AB BC 2. 3. BD BD 3. 4. BD is a perpendicular bisector and an angle bisector. 4. Altitudes coincide with perpendicular bisectors and angle bisectors in triangles. 5. AD DC 5. 6. BD AC 6. 7. BDA and BDC are right angles 7. 8. 8. All right angles are congruent. 9. ABD CBD 9. 10. A C 10. 11. ABD CBD. 12. All corresponding angles are congruent and all corresponding sides are congruent.
5-4 Homework 1) 2) 3) Find x and y in the following traingle, given that triangle MNP is congruent to triangle TUS 4. Given: CD bisects AB. Prove: AE = EB
5. Given: AD is a perpendicular bisector and AD is an angle bisector. ΔABC is isosceles. Prove that triangle ADC is congruent to triangle ADB. Fill in: Here we showed that because all sides and all corresponding are congruent triangle ADC is congruent to triangle ADB