6.1 Warmup Per ate Instructions: Justify each statement using your Vocab/Theorems ook. If!! =!! and!! = 50, then!! = 50. P F S If!" is rotated 180 around point F, then!"!" If!!"# +!!"# = 180, then!"# is supplementary to!"#. P F S If!"!", then!"!". If!!"# +!!"# = 180 and!!"# = 80, then!!"# + 80 = 180. If!! = 20 and!! = 20, then!! Z X Y If!!"# = 90 and!!"# = 90, then!"#!"#. Page 1
6.2 Proof Strategies Per ate Proof Writing Strategies proof is a logical string of statements and reasons designed to convince someone of a conclusion. ll proofs begin with something true. These can either be statements given in a diagram, definitions, axioms or previously proved theorems. Here are some basic rules to follow when writing proofs. 1- etermine exactly what you are trying to prove. If needed, write it in terms of specific angles or sides, not just the angles are congruent. 2- Make sure your steps lead logically from one to the next. 3- Never assume. If you make a statement, you must have a reason. Use what is given and your Vocab/Theorems ooklets. elow is an example of a simple statement that can be proven by connecting given facts together to reach a conclusion. Problem 1 reate a flowchart proof for the following statement: Given:!! and!! = 25 Prove:!! = 25 s s The statements and justifications have been mixed up below. opy them into the boxes above in a logical order to complete the proof. Hint: You should begin with things that are given and end with what you want to prove.!!!! = 25 Given!! = 25!! =!! ef. ngle ongruence Given Substitution Page 2
6.2 Proof Strategies Per ate Flowchart Proofs: The flowchart proof on the previous page was sequential: each step led logically to the next one. However, in geometry we often do not have a simple straight line of facts-to-conclusion. There are branches without a set order. Example 2 m! = 150.00 Given:!!"# = 150 and!"# is a straight angle. Prove:!!"# = 30. #1!!"# = 150 Given #2!"# is adjacent to!"# Given (by diagram) #3!"# is a straight angle Given #4!!"# +!!"# =!!"# ngle ddition Postulate, 2 #5!!"# = 180 ef. Straight ngle, 3 #6 150 +!!"# = 180 Substitution, 1, 4, 5 onclusion #7!!"# = 30 Subtraction Property, 6 Page 3
6.2 Proof Strategies Per ate Example 2 - ontinued Information given in the diagram does not need a reason other than given to justify its place in a proof. However, all other statements need a justification connecting them to the given information. nother way of organizing these statements is by creating a two-column proof, which was introduced in the previous lesson. elow is an example of the flowchart proof above translated into a two-column proof. s 1. m! =150 2.! is adjacent to! 3.! is a straight angle 4. m! + m! = m! 5. m! =180 6. 150 + m! =180 7. m! = 30 s 1. 2. 3. 4. 5. 6. 7. 4 and 5 must be placed in order in the two-column proof but are on the same level in the flowchart proof. This is a limitation of the two-column proof. The order can be slightly different but still result in a valid proof. In general, the given statements are best placed at the beginning and the conclusion at the end, but what is in the middle of you proof may vary. s Fill in the missing justifications in the table above using the options given in the list to the right. ll the needed reasons are listed, but they are not necessarily in the correct order. Given Subtraction Property, 6 ngle ddition Postulate, 2 Given ef. Straight ngle, 3 Given Substitution, 1, 4, 5 Page 4
6.2 Proof Strategies Per ate Example 3 reate a flowchart proof for the situation below. hoose your statements from the list and number them as you go. Fill in the missing s for those statements using your Vocab/Theorems ooklets. 1 2 3 Given: 1 and 2 are complementary. 2 and 3 are complementary. Prove: 1 3. # Given # ef. omplementary # # s Note: Not in order! 1 +! 2 =! 2 +! 3 1 and 2 are complementary! 1 +! 2 = 90 2 and 3 are complementary 1 3! 1 =! 3! 2 +! 3 = 90 # # onclusion. # Page 5
6.2 Proof Strategies Per ate Example 3 ontinued Use your flowchart proof from the previous page to create a two-column proof in the table below. s s Now compare your proof with your table partner. Help each other answer the questions below. Reflection 1. id your two-column proof have the same order as your table partner? Why or why not? 2. To what extend does the order matter in a two-column proof? Would the proof still be valid if you arbitrarily rearranged your statements? Page 6
6.2 Proof Strategies Per ate Example 4 Prove the following conjecture using a flowchart proof and then check your answer by filling out the missing parts of the two-column proof on the next page. Some statements that you will be using are listed in the table. Note: Not all of them are given. You will have to come up with the rest on your own. Use what is given in the diagram and definitions and axioms in your Vocab/Theorems ook. Some Possible s E 1 2 3!!"# =! 1 +! 2! 1 =! 3!!"# =!!"# Some Possible s djacent ngle Theorem Substitution Given:! 1 =! 3 Prove:!"#!"# Write the flowchart proof in the space below: Page 7
6.2 Proof Strategies Per ate Example 4 ontinued Fill in the missing parts of the proof. Use your flowchart proof from the previous page. 1.! 1 =! 3 1. Given 2.!!"# =! 1 +! 2 3. 4.!!"# =! 3 +! 2 5. 6.!"#!"# 2. 3. 4. 5. Substitution, 3, 4 6. Now share your proof with your table partner and answer the reflection questions together. 1. What additional s did you use in this proof? 2. o you think there are any other ways to prove this statement? If so, how would it be done? 3. Which method of proof do you find easier to create/understand, two-column or flowchart? Why? Page 8
6.2 Proof Strategies Per ate Vertical ngle Theorem (#THM) We will now prove one of the major theorems of this unit, the Vertical ngle Theorem. It states: Vertical angles are congruent. E Review prior knowledge: Name both pairs of angles in the diagram that are vertical. and and Now let s prove the theorem so that we can use it in future proofs. We will begin by writing the statement more formally in reference to the diagram above. We are trying to show that vertical angles are congruent based on the diagram above. Fill out the Given statements and what you want to Prove in the space below. Please be specific. Name the angles and lines in reference to the diagram. Given: Prove: iscuss your given statements and what you want to prove with your table partner. id you word them differently? re your statements equivalent? Page 9
6.2 Proof Strategies Per ate The Vertical ngle Theorem Proof Prove the following theorem. E Given: Two lines,!" and!", intersect at point. Prove:!"#!"# You may use a flowchart proof to organize your work if needed. Page 10
6.3 - Parallel Lines Per ate Parallel Lines Postulate We will now explore parallel lines and prove some angle congruencies. G H E F Review prior knowledge:!"!" and!" is a transversal. Match all the angles in the diagram that appear congruent to the angles listed below. This is clearly not a proof but we have seen these angles before and intuitively some of them seem like they could be congruent. Questions: 1.!"# and and 2.!"# and and 3. If!!"# = 27, find the following: a.!!"# = b.!!"# = c.!!"# = d.!!"# = e.!!"# +!!"# = Page 11
6.3 - Parallel Lines Per ate Parallel Lines Proofs Euclid s 5 th axiom states: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. This is equivalent to the following postulate, which we will be exploring: If parallel lines are cut by a transversal, then the two interior angles on the same side of the transversal sum to 180 degrees. G H E F This postulate says something specific about the sum of several pairs of angles in the diagram above. We will use it to prove many theorems in this unit. 1. Write two equations that follow directly from the parallel lines postulate. 2. Use the parallel lines postulate and the diagram above to complete the following proof. Given: Line!"!" Prove:!!"# +!!"# =!!"# +!!"# 1. 2.!!"# +!!"# = 180 3.!!"# +!!"# = 180 4.!!"# +!!"# =!!"# +!!"# 1. Given 2., 1 3., 1 4., 2, 3 Page 12
6.3 - Parallel Lines Per ate Practice 1 - orresponding ngles re ongruent (#THM) In the figure below, angle 1 and angle 2 are corresponding. If the lines are parallel, we can show that corresponding angles are equal to each other. Prove the statement using a two-column proof. You may first organize your ideas in a flowchart proof if desired. 2 n 3 1 m q Given: Lines m and n are parallel, and q is a transversal. Prove: 1 2 Page 13
6.3 - Parallel Lines Per ate Practice 2 - lternate Interior ngles are ongruent (#THM) In the figure below there are two sets of alternating interior angles, one pair of which has been highlighted. If the lines are parallel, we can show that these angles are congruent. Prove the statement using a two-column proof. G H E F Given:!"!" and!" is a transversal. Prove:!"#!"# Page 14
6.3 - Parallel Lines Per ate Practice 3 - lternate Exterior ngles are ongruent (#THM) In the figure below there are two sets of alternating exterior angles, one pair of which has been highlighted. If the lines are parallel, we can prove that these angles are congruent. Prove the statement using a two-column proof. G H E F Given:!"!" and!" is a transversal. Prove:!"#!"#. Page 15
6.4 Homework Per ate Instructions: Prove each statement below using a two-column proof. 1. 3 4 1 2 Given:!"!" and! 1 =! 2. Prove:! 3 =! 4 Page 16
6.4 Homework Per ate 2. G H E F Given:!"!" and!" is a transversal. Prove:!"# is supplementary to!"#. Page 17
6.4 Homework Per ate 3. G H E F Given:!"!" and!" is a transversal. Prove:!!"# +!!"# +!!"# +!!"# = 360 Page 18