Pre-AP Geometry 1

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Pre-AP Geometry 1

• Unit 2 Deductive Reasoning

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Pre-AP Geometry 1 Unit 2

• 2.1 If-then statements, converse, and
  biconditional statements

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Conditional Statements

• Conditional Statement-
• A statement with two parts (hypothesis and
  conclusion)
• Also known as Conditionals
• If-then form
• A way of writing a conditional statement that
  clearly showcases the hypothesis and conclusion
  p?q
• Hypothesis-
• The if part of a conditional statement
• Represented by the letter p
• Conclusion
• The then part of a conditional statement
• Represented by the letter q

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Conditional Statements

• Examples of Conditional Statements
• If today is Saturday, then tomorrow is Sunday.
• If its a triangle, then it has a right angle.
• If x2 4, then x 2.
• If you clean your room, then you can go to the
  mall.
• If p, then q.

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Conditional Statements

• Example 1
• Circle the hypothesis and underline the
  conclusion in each conditional statement
• If you are in Geometry 1, then you will learn
  about the building blocks of geometry
• If two points lie on the same line, then they are
  collinear
• If a figure is a plane, then it is defined by 3
  distinct points

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Conditional Statements

• Example 2
• Rewrite each statement in ifthen form
• A line contains at least two points
• When two planes intersect their intersection is a
  line
• Two angles that add to 90 are complementary

If a figure is a line, then it contains at least
two points

• If two planes intersect, then their intersection
  is a line.

If two angles add to equal 90, then they are
complementary.
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Conditional Statements

• Counterexample
• An example that proves that a given statement is
  false
• Write a counterexample
• If x2 9, then x 3

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Conditional Statements

• Example 3
• Determine if the following statements are true or
  false.
• If false, give a counterexample.
• If x 1 0, then x -1
• If a polygon has six sides, then it is a decagon.
• If the angles are a linear pair, then the sum of
  the measure of the angles is 90º.

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Conditional Statements

• Converse
• Formed by switching the if and the then part.
• Original
• If you like green, then you will love my new
  shirt.
• Converse
• If you love my new shirt, then you like green.

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Biconditional Statements

• Can be rewritten with If and only if
• Only occurs when the statement and the converse
  of the statement are both true.
• A biconditional can be split into a conditional
  and its converse.
• p if and only if q
• All definitions can be written as biconditional
  statements

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Example

• Give the converse of the statement.
• If the converse and the statement are both true,
  then rewrite as a biconditional statement
• If it is Thanksgiving, then there is no school.
• If an angle measures 90º, then it is a right
  angle.

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Quiz- Get out a piece of paper and answer the
following questions

• Underline the hypothesis and circle the
  conclusion. Then, write the converse of the
  statement. If the converse and the statement are
  true, rewrite as a biconditional statement. If
  not, give a counterexample.
• 1. If a number is divisible by 10, then it is
  divisible by 5.
• 2. If today is Friday, then tomorrow is Saturday.
• 3. If segment DE is congruent to segment EF, then
  E is the midpoint of segment DF.

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Assignment

• Lesson 2.1
• P. 35 2-30 even

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Pre-AP Geometry 1 Unit 2

• 2.2 Properties from Algebra
• p. 37

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Properties of equality

• Addition property
• If a b, then a c b c
• Subtraction property
• If a b, then a c b c
• Multiplication property
• If a b, then ac bc
• Division property
• If a b, then

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Reasoning with Properties from Algebra

• Reflexive property
• For any real number a, a a
• Symmetric property
• If ab, then b a
• If
• Transitive Property
• If a b and b c, then a c
• If ?D ?E and ?E ?F, then ?D ?F
• Substitution property
• If a b, then a can be substituted for b in any
  equation or expression
• Distributive property
• 2(x y) 2x 2y

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Two-column proof

• A way of organizing a proof in which the
  statements are made in the left column and the
  reasons (justification) is in the right column
• Given Information that is given as fact in the
  problem.

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Reasoning with Properties from Algebra

• Example 1
• Solve 6x 5 2x 3 and write a reason for each
  step

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Reasoning with Properties from Algebra

• Example 2
• 2(x 3) 6x 6

• Given

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Reasoning with Properties from Algebra

• Determine if the equations are valid or invalid,
  and state which algebraic property is applied
• (x 2)(x 2) x2 4
• x3x3 x6
• -(x y) x y

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Warmup

• With a partner, Complete proof 11 and 12 on p.
  40

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Proving Theorems

• Lesson 2.3
• Pre-AP Geometry

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Proofs

• Geometric proof is deductive reasoning at work.
• Throughout a deductive proof, the statements
  that are made are specific examples of more
  general situations, as is explained in the
  "reasons" column.
• Recall, a theorem is a statement that can be
  proved.

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Vocabulary

• Definition of a Midpoint
• The point that divides, or bisects, a segment
  into two congruent segments.
• If M is the midpoint of AB, then AM is congruent
  to MB
• Bisect
• To divide into two congruent parts.
• Segment Bisector
• A segment, line, or plane that intersects a
  segment at its midpoint.

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Midpoint Theorem
If M is the midpoint of AB, then AM ½AB and MB
½AB
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Proof Midpoint Formula
Given M is the midpoint of Segment AB Prove
AM ½AB MB ½AB
Statement 1. M is the midpoints of segment
AB 2. Segment AM Segment MB, or AM MB
3. AM MB AB 4. AM AM AB, or 2AM AB
5. AM ½AB  6. MB ½AB
Reason 1. Given 2. Definition of midpoint 3.
Segment Addition Postulate 4. Substitution
Property (Steps 2 and 3) 5. Division
Prop. of  Equality 6. Substitution Property.
(Steps 2 and 5)
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The Midpoint Formula

• The Midpoint Formula
• If A(x1, y1) and B(x2, y2) are points in a
  coordinate plane, then the midpoint of segment AB
  has coordinates

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The Midpoint Formula

• Application
• Find the midpoint of the segment defined by the
  points A(5, 4) and B(-3, 2).

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Midpoint Formula

• Application
• Find the coordinates of the other endpoint B(x,
  y) of a segment with endpoint C(3, 0) and
  midpoint M(3, 4).

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Vocabulary

• Definition of an Angle Bisector
• A ray that divides an angle into two adjacent
  angles that are congruent.
• If Ray BD bisects angle ABC, then ABD is
  congruent to DBC

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Angle Bisector Theorem

• If BX is the bisector of ?ABC, then the measure
  of ?ABX is one half the measure of ?ABC
  and the measure of ?XBC one half of the ?ABC.

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Proof Angle Bisector Theorem

• Given BX is the bisector of ?ABC.
• Prove m ?ABX ½ m ?ABC m ?XBC ½m ?ABC

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Reasons used in proofs

• Given
• Definitions
• Postulates
• Theorems

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2.4 Special Pairs of Angles

• Page 50
• Pre-AP Geometry 1

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Angle Pair Relationships

• Complementary Angles
• Two angles that have a sum of 90º
• Each angle is a complement of the other.
• Non-adjacent complementary Adjacent
  angles complementary angles

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Angle Pair Relationships

• Supplementary Angles
• Two angles that have a sum of 180º
• Each angle is a supplement of the other.

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Angle Pair Relationships

• Example 1
• Given that ?G is a supplement of ?H and m?G is
  82, find m?H.
• Given that ?U is a complement of ?V, and m?U is
  73, find m?V.

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Angle Pair Relationships

• Example 2
• ?T and ?S are supplementary.
• The measure of ?T is half the measure of ?S.
  Find m?S.

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Angle Pair Relationships

• Example 3
• ?D and ?E are complements and ?D and ?F are
  supplements. If m?E is four times m?D, find the
  measure of each of the three angles.

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Theorem 2-3

• Vertical angles are congruent
• Given angle 1 and angle 2 are vertical angles
• Prove?1? ?2

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2
1
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Angle pair relationships

• Find x and the measure of each angle.

?A
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2x 10
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2.5 Perpendicular Lines

• Page 56
• Pre-AP Geometry 1

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Perpendicular lines

• Two lines that intersect to form right angles
• We use the symbol ? to show that lines are
  perpendicular. Line AB ? Line CD

C
A
B
D
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Perpendicular lines theorems

• Theorem 2-4 If two lines are perpendicular, then
  they form congruent adjacent angles
• Theorem 2-5 If two lines form congruent adjacent
  angles, then the lines are perpendicular
• Theorem 2-6 If the exterior sides of two
  adjacent angles are perpendicular, then the
  angles are complementary.

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Unit 2.6 Planning a proof

• p. 60
• Pre-AP Geometry 1
• September 11, 2008

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Parts of a proof

• Statement of the theorem you are trying to prove
• A diagram to illustrate given information
• A list of the given information
• A list of what you are trying to prove
• A series of Statements and Reasons that lead from
  the given information to what you are trying to
  prove.

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Example proof of theorem 2-7

• If 2 angles are supplements of congruent angles,
  then the two angles are congruent.
• Given ?2 ? ?4
• ?1 and ?2 are supplementary
• ?3 and ?4 are supplementary
• Prove ?1 ? ?3

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Theorem 2-8

• If two angles are complements of congruent
  angles, then the two angles are congruent.
• Prove theorem 2-8. Use the proof from theorem 2-7
  (p. 61) to help. You may do this with a partner.
  Due at end of hour. Make sure you include all 5
  parts (p. 60).