2.6 Prove Statements about Segments and Angles 2.7 Prove Angle Pair Relationships

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2.6 Prove Statements about Segments and
Angles2.7 Prove Angle Pair Relationships

• Objectives
• To write proofs using geometric theorems
• To use and prove properties of special pairs of
  angles to find angle measurements

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Thanks a lot, Euclid!

• Recall that it was the development of
  civilization in general and specifically a series
  of clever ancient Greeks who are to be thanked
  (or blamed) for the insistence on reason and
  proof in mathematics.

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Premises in Geometric Arguments

• The following is a list of premises that can be
  used in geometric proofs
• Definitions and undefined terms
• Properties of algebra, equality, and congruence
• Postulates of geometry
• Previously accepted or proven geometric
  conjectures (theorems)

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Properties of Equality

• Maybe you remember these from Algebra.

Reflexive Property of Equality For any real number a, a a.
Symmetric Property of Equality For any real numbers a and b, if a b, then b a.
Transitive Property of Equality For any real numbers a, b, and c, if a b and b c, then a c.
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Theorems of Congruence

• Congruence of Segments
• Segment congruence is reflexive, symmetric, and
  transitive.

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Theorems of Congruence

• Congruence of Angles
• Angle congruence is reflexive, symmetric, and
  transitive.

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Example 1a

• Given
• Prove

Statements Reasons




1.Given
1.
2. has length AB
2.Ruler Postulate
3.Reflexive Prop. of
3. AB AB
4.
4.Definition of Congruent Segments
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Example 1b
Given Prove
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Example 2

• Prove the following
• If M is the midpoint of AB, then AB is twice AM
  and AM is one half of AB.

Prove AB 2AM and AM (1/2)AB
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Example 3a

• If there was a right angle in Denton, TX, and
  other right angle in that place in Greece with
  all the ruins (Athens), what would be true about
  their measures?

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Right Angle Congruence Theorem

• All right angles are congruent.
• Yes, it seems obvious, but can you prove it?
  What would be your Given information? What would
  you have to prove?

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Example 3b
Given lt A and lt B are right angles Prove
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Linear Pair Postulate

• If two angles form a linear pair, then they are
  supplementary.

Do we have to prove this?
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Example 4

• Given
• Prove

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Congruent Supplements

• Suppose your angles were numbered as shown.
  Notice angles 1 and 2 are supplementary. Notice
  also that 2 and 3 are supplementary. What must
  be true about angles 1 and 3?

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Congruent Supplement Theorem

• If two angles are supplementary to the same angle
  (or to congruent angles), then they are congruent.

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Example 5

• Prove the Congruent Supplement Theorem.

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What to Prove

• Notice that you can essentially have two kinds of
  proofs
• Proof of the Theorem
• Someone has already proven this. You are just
  showing your peerless deductive skills to prove
  it, too.
• YOU CANNOT USE THE THEOREM TO PROVE THE THEOREM!
• Proof Using the Theorem (or Postulate)

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Congruent Complement Theorem

• If two angles are complementary to the same angle
  (or to congruent angles), then they are congruent.

Youll have to prove this in your homework.
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Vertical Angle Congruence Theorem

• Vertical angles are congruent.

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Example 6

• Prove the Vertical Angles Congruence Theorem.

Given lt 1 and lt 3 are vertical angles
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Example 7

• Given
• Prove

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Example 8

• Given
• Prove lt 3 and lt 4 are supplements

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Example 9