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Conditional Statements
Lesson 2.1 Conditional Statements You will learn to… * recognize and analyze a conditional statement * write postulates about points, lines, and planes using conditional statements
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A conditional statement has two parts, a hypothesis and a conclusion.
p q If p, then q.
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hypothesis (p) conclusion (q) If the team wins the game,
then they will win the tournament. conclusion (q)
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If two planes intersect, then their intersection is a line.
Write an if-then statement. 1. The intersection of two planes is a line. If two planes intersect, then their intersection is a line.
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Write an if-then statement.
2. A line containing two given points lies in a plane if the two points lie in the plane. If two points lie in a plane, then the line containing them lies in the plane.
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The negation of a statement is formed by negating the statement.
The negation is written ~ p.
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4. m A = 125° m A 125° 5. A is not obtuse A is obtuse
Write the negation of this statement. 4. m A = 125° m A 125° 5. A is not obtuse A is obtuse
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The inverse is formed by negating the hypothesis and the conclusion.
The inverse is ~ p ~ q. If ~ p, then ~ q.
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If m A 125°, then A is not obtuse.
Write the inverse of this if-then statement. Is it true or false? 6. If m A = 125°, then A is obtuse. If m A 125°, then A is not obtuse. False
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The converse is formed by switching the hypothesis and conclusion.
The converse is q p. If q, then p.
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If A is obtuse, then m A = 125°.
Write the converse of this if-then statement. Is it true or false? 3. If m A = 125°, then A is obtuse. If A is obtuse, then m A = 125°. False
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The contrapositive is ~ q ~ p.
The contrapositive is formed by negating the hypothesis and conclusion of the converse. The contrapositive is ~ q ~ p. If ~ q, then ~ p.
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If A is not obtuse, then m A 125°.
Write the contrapositive of this if-then statement. Is it true or false? 7. If m A = 125°, then A is obtuse. If A is not obtuse, then m A 125°. True
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Through any two points there exists exactly one line.
Postulate 5 Through any two points there exists exactly one line.
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Postulate 6 A line contains at least two points.
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If two lines intersect, then their intersection is exactly one point.
Postulate 7 If two lines intersect, then their intersection is exactly one point.
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Through any three noncollinear points there exists exactly one plane.
Postulate 8 Through any three noncollinear points there exists exactly one plane. T B A C
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A plane contains at least three noncollinear points.
Postulate 9 A plane contains at least three noncollinear points.
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Postulate 10 If two points lie in a plane, then the line containing them lies in the plane.
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If 2 planes intersect, then their intersection is ___________.
Postulate 11 If 2 planes intersect, then their intersection is ___________. a line
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Biconditional Statements
Lesson 2.2 Biconditional Statements You will learn to… * recognize and use definitions * recognize and use biconditional statements
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All definitions can be interpreted “forward” and “backward.”
All definitions are biconditional.
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two lines that intersect
For example, perpendicular lines are defined as two lines that intersect to form one right angle.
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If two lines are perpendicular, then they intersect to form one right angle.
If two lines intersect to form one right angle, then they are perpendicular.
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A biconditional statement contains the phrase “if and only if.”
Two lines are perpendicular if and only if they intersect to form one right angle.
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A biconditional statement is true when the original if-then statement
AND its converse are both true.
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1. Two angles are supplementary
if and only if the sum of their measures is 180°. if-then statement: If two angles are supplementary, then the sum of their measures is 180°. converse: If the sum of the measures of two angles is 180°, then they are supplementary.
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2. If an angle is 135˚, then it is an obtuse angle.
converse: If an angle is obtuse, then its measure is 135°. Can we write a biconditional statement? counterexample?
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3. If two angle measures add up to 90˚, then they are complementary angles.
converse: If two angles are complementary, then the sum of their measures is 90°. Can we write a biconditional statement? Two angles are complementary if and only if the sum of their measures is 90°.
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Workbook Page 25 (1-7)
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Deductive Reasoning Lesson 2.3 You will learn to…
* use symbolic notation to represent logical statements * form conclusions by applying laws of logic
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p: mB = 90˚ q: B is a right angle
Using these phrases, write the conditional statement. p: mB = 90˚ q: B is a right angle If mB = 90˚, then B is a right angle. 1. p q If B is a right angle, then mB = 90˚ If mB ≠ 90˚, then B is not a right angle. 2. q p If B is not a right angle, then mB ≠ 90˚ 3. ~ p ~ q mB = 90˚ if and only if B is a right angle. 4. ~ q ~ p 5. p q
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Deductive Reasoning uses facts to make a logical argument.
definitions, properties, postulates, theorems, and laws of logic
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Law of Detachment p q p q Given facts Therefore:
hypothesis is true q conclusion must be true Therefore: You can use these symbols when asked to explain your reasoning.
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Law of Detachment If I learn my facts, then I will pass geometry. I learned my facts. q p p Therefore, I passed geometry. q
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Law of Syllogism p q q r p r Given facts Therefore:
You can use these symbols when asked to explain your reasoning.
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Law of Syllogism If I pass geometry, then my dad will be happy. If my dad is happy, then I will get a cell phone. p q q r Therefore, if I pass geometry, then I will get a cell phone. p r
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6. Is this argument valid? If 2 lines in a plane are parallel, then they do not intersect. p q Coplanar lines n and m are parallel. p Therefore, lines n and m do not intersect. q VALID – Law of Detachment
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VALID – Law of Syllogism
7. Is this argument valid? If 2 angles are supplementary, then the sum of their measures is 180˚. p q r p p q If 2 angles form a linear pair, then they are supplementary. r p Therefore, if 2 angles form a linear pair, then the sum of their measures is 180˚ r q VALID – Law of Syllogism
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8. Is this argument valid? p q q p INVALID
If 2 angles are a linear pair, then the sum of their measures is 180˚. p q m1 + m2 = 180˚ q Therefore, 1 and 2 are a linear pair. p INVALID
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9. Is this argument valid? r q p q r q p q p r INVALID
If you live in Canada, then you live in North America. r q p q p q If you live in South Carolina, then you live in North America. r q Therefore, if you live in Canada, then you live in South Carolina p r INVALID
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If you use this product, then you will have even-toned skin. If you have even-toned skin, then If you use this product, then you will be beautiful. you will be beautiful.
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Properties of Equality and Congruence
Lesson 2.4 Properties of Equality and Congruence You will learn to… * use properties from algebra * use properties of length and measure to justify segment and angle relationships
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Reflexive Property Symmetric Property Transitive Property
Equality Properties Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property
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Reflexive Property Mrs. R is the same height as Mrs. R. A = A
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Symmetric Property If A = B , then B = A Mr. S is the same
height as Mrs. R THEN IF Mrs. R is the same height as Mr. S If A = B , then B = A
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Transitive Property If A = B and B = C, then A = C Mr. S is the
same height as Mrs. T AND Mrs. R is the same height as Mrs. T THEN IF Mrs. R is the same height as Mr. S If A = B and B = C, then A = C
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Reflexive Property
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Symmetric Property
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Transitive Property If XY = ST and ST = 10, then XY = 10 If mA = mB and mB = 10°, then mA = 10°
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Division Property If 8x=16, then x=2.
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Addition Property If x-7=5, then x=12.
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Multiplication Property
If ½ x = 7, then x=14.
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Subtraction Property If x+3=7, then x=4.
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Substitution Property
If A=x2 and x=6, then A=36. If 4 + 7x – 10 = 24, Then 7x - 6 = 24
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Distributive Property
If B=2(4x + 3), then B=8x + 6. If 4x + 7x = 24, Then 11x = 24
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Memorize definitions, postulates, and theorems as we learn them.
Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!! Let’s Practice…
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2. 4+2(3x+5)=11-x Given Distributive prop. 4+6x+10=11-x 14+6x =11-x
Substitution 14 + 7x = 11 Addition prop. 7x = - 3 Subtraction prop. x = - 3/7 Division Prop.
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4. Given 1/5 x + 4 = 2x + 3/5 1x + 20 = 10x + 3 Multiplication Prop
Subtraction Prop Subtraction Prop 17 = 9x 17/9 = X Division Prop
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5. Given that MN=PQ, show that MP=NQ M N P Q Given MN = PQ
Segment Addition Postulate MP = MN + NP MP = PQ + NP Substitution Prop Segment Addition Postulate NQ = PQ + NP MP = NQ Substitution Prop
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7. Given mAQB=mCQD,show that mAQC=mBQD
Angle Addition Postulate mAQB + mBQC = mAQC mCQD + mBQC = mAQC Substitution Angle Addition Postulate mCQD + mBQC = mBQD mAQC = mBQD Substitution
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show that mRPV=3(mRPS)
8. Given mRPS=mTPV and mTPV=mSPT show that mRPV=3(mRPS) P R S T V mRPS = mTPV Given mTPV = mSPT Given mRPS = mSPT Transitive Prop mRPV= mRPS+mSPT+mTPV Angle Addition Postulate mRPV= mRPS+mRPS+mRPS Substitution mRPV = 3(mRPS) Distributive Prop
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You can use definitions
as reasons in proofs. Statements Reasons 1) A is a right angle 1) Given 2) m A = 90˚ 2) Def. of right angles
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Statements Reasons 1) Given 1) m A = 90˚ 2) A is a right angle 2)
Def. of right angles
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Statements Reasons 1) AB CD 1) Given 2) 1 is a right angle 2)
Def. of lines
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Statements Reasons 1 is a right angle 1) Given 2) AB CD 2)
Def. of lines
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1 and 2 are vertical angles 1)
Statements Reasons 1 and 2 are vertical angles 1) Def. of vertical angles Vertical Angles Theorem 2) 1 2 2)
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2) 1 & 2 are supplementary 2)
Statements Reasons 1 and 2 are a linear pair 1) Def. of linear pair 2) 1 & 2 are supplementary 2) Linear Pair Postulate Def. of supplementary 3) m1 + m = 180° 3)
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Statements Reasons 1) AB = CD 1) Given 2) AB CD 2) Def. of segment
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Statements Reasons 1) Given 1) AB CD 2) AB = CD 2) Def. of segment
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Statements Reasons 1) m 1 = m 2 1) Given 2) 1 2 2)
Def. of angles
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Statements Reasons 1) 1 2 1) Given 2) m 1 = m 2 2) Def. of
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Proving Statements about Segments
Lesson 2.5 Proving Statements about Segments You will learn to… * justify statements about congruent segments * write reasons for steps in a proof use practice sheet of proofs
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Memorize definitions, postulates, and theorems as we learn them.
Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!!
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Reflexive Property of Congruence
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Symmetric Property of Congruence
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Transitive Property of Congruence
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(Proof is on next slide)
1. Given: EF = GH Prove: EG FH H G F E (Proof is on next slide)
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1. Given Addition Prop. Statements Reasons 1) EF = GH 1)
2) EF + FG = GH + FG 2) 3) EG = EF + FG FH = GH + FG 3) 4) EG = FH 4) 5) EG FH 5) Given Addition Prop. Segment Addition Postulate Substitution Def. of
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2. Given: RT WY, ST = WX Prove: RS XY R S T Y X W
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2. Given Given Statements Reasons 1) RT WY 1) 2) RT = WY 2)
3) RT = RS + ST WY = WX + XY 3) 4) RS + ST = WX + XY 4) 5) ST = WX 5) 6) RS + ST = ST + XY 6) 7) RS = XY 7) 8) RS XY 8) Given Def. of Segment Addition Postulate Substitution Given Substitution Subtraction Prop. Def. of
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3. Given: X is the midpoint of MN
and MX = RX Prove: XN = RX R X S N M
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3. Statements Reasons 1) X is the midpoint of MN 1) 2) NX = MX 2)
3) MX = RX 3) 4) NX = RX 4) Given Def. of midpoint Given Transitive Prop.
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Paragraph proof example for #1
Since EF = GH, EF + FG = GH + FG by the Addition Property. EG = EF + FG and FH = GH + FG by the Segment Addition Postulate. By Substitution, EG = FH. Therefore, EG FH by the definition of congruent segments.
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Paragraph proof example for #3
So, I was chillin’ with the homeboys and my homeboy Sherrod tells me, “Dave, x is the midpoint of MN, so NX = MX.” I said, “Sherrod, how do you figure?” Sherrod tells me “The definition of midpoint says so!” So I was like, “yo, Sherrod, did you know MX = RX,” and he said, “really, well then NX = RX Dawg. “Sherrod, my homie, I didn’t know you were so smart,” I said, “how did you figure that out?” He was like, “Substitution, my brother!” David Mathews
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# 17 Statements 1) XY = 8, XZ = 8, 2) XY = XZ 3) XY XZ 4) XY ZY 5) XZ ZY
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# 18 Statements 1) NK NL, NK = 13 2) NK = NL 3) NL = 13
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Proving Statements about Angles
Lesson 2.6 Proving Statements about Angles You will learn to… * use angle congruence * prove properties about special pairs of angles
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All ________ angles are __________. congruent
Theorem 2.3 Right Angle Congruence Theorem right All ________ angles are __________. congruent
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A is supplementary to 40° B is supplementary to 40° What do you know about A and B?
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Theorem 2.4 Congruent Supplements Theorem
If 2 angles are supplementary to the same angle, then they are _______________. congruent
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Using the Congruent Supplements Theorem…
Reasons Statements 1 & 2 are supp. 1 & 3 are supp. 1) 2) 2 3 2) Congruent Supplements Theorem
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A is complementary to 50° B is complementary to 50° What do you know about A and B?
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Theorem 2.5 Congruent Complements Theorem
If 2 angles are complementary to the same angle, then they are _______________. congruent
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Using the Congruent Complements Theorem…
Reasons Statements 1 & 2 are comp. 1 & 3 are comp. 1) 2) 2 3 2) Congruent Compliments Theorem
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If two angles form a linear pair, then they are _______________.
Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are _______________. supplementary
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Using the Linear Pair Postulate…
Statements Reasons 1 & 2 are a linear pair 1) 1) Def. of linear pair 2) 1 & 2 are supplementary 2) Linear Pair Postulate 3) m1 + m2 = 180 3) Def. of supplementary
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Vertical angles are _______________.
Theorem 2.6 Vertical Angles Theorem Vertical angles are _______________. congruent
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Using the Vertical Angles Theorem…
Statements Reasons 1) 1) Def. of vertical angles 1 & 2 are vertical angles 2) 1 2 2) Vertical Angles Theorem
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1. Given: 1 2 , 3 4 , 2 3 Prove: 1 4 3 1 2 4
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1. Statements Reasons 1. 1 2 , 2 3 1. 2. 1 3 2. 3. 3 4
1. 1 2 , 2 3 1. 2. 1 3 2. 3. 3 4 3. 4. 1 4 4. Given Transitive Prop. Given Transitive Prop.
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2. Given: m1 = 63˚,1 3 , 3 4 Prove: m4 = 63˚ 1 2 3 4
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2. Statements Reasons 1. m1 = 63˚, 1 3 , 3 4 1. 2. 1 4 2.
2. 1 4 2. 3. m1 = m4 3. 4. m4 = 63˚ 4. Given Transitive Prop. Def of Substitution
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3. Given: DAB & ABC are right angles , ABC BCD
Prove: DAB BCD D C A B
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3. Given Given Transitive Prop. Statements Reasons 1. DAB & ABC
are right angles 1. 2. DAB ABC 2. 3. ABC BCD 3. 4. DAB BCD 4. Given All right s are Given Transitive Prop.
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4. Given: m1 = 24˚,m3 = 24˚ 1 & 2 are complementary 3 & 4 are complementary
Prove: 2 4 1 2 3 4
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4. Statements Reasons m1 = 24˚, m3 = 24˚
1. 1 & 2 are comp. 3 & 4 are comp. 1. 2. m1 = m3 2. 3. 1 3 3. 4. 2 4 4. Given Substitution Def of Congruent Complements Theorem
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5. Statements Reasons 1. 1 and 2 are a linear pair 2 and 3 are 1.
2. 1 and 2 are supp. 2 and 3 are supp. 2. 3. 1 3 3. Given Linear Pair Postulate Congruent Supplements Theorem
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6. Statements Reasons 1. QVW and RWV are supplementary 1.
2. QVW and QVP are a linear pair 2. 3. QVW and QVP are supplementary 3. 4. QVP RWV 4. Given Def. of Linear Pair Linear Pair Postulate Congruent Supplements Theorem
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#24 & #26 for homework #24 Statements 1) 3 and 2 are complementary
2) m1 + m2 = 90 3) m 3 + m2 = 90 4) m1 + m2 = m3 + m2 5) m1 = m3 6) 1 3
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#26 Statements 1) 4 and 5 are vertical angles 2) 6 and 7 are vertical angles 3) 4 5 , 6 7 4) 5 6 5) 4 7
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