2.6 Proving Statements about Angles

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2.6 Proving Statements about Angles

• p. 109

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Theorem 2.2 Properties of Angle Congruence

• Angle congruence is reflexive, symmetric, and
  transitive.

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Proof of transitive partGiven ltA ltB and ltB
ltCProve ltA ltC

• Statements
• ltA ltB ltB ltC
• mltAmltB mltBmltC
• mltAmltC
• ltA ltC
• You will be proving the reflexive symmetric
  parts for homework.

• Reasons
• Given
• Defn. of lts
• Trans. Prop. Of
• Defn. of lts

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Thm 2.3 Right Angle Congruence Thm.
All right lts are .

• Statements
• ltA ltB are right lts.
• mltA90o mltB90o
• mltAmltB
• ltA ltB

• Reasons
• Given
• Defn. of rt. lt
• Subst. prop of
• Defn. of lts

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Thm 2.4 - supplements thm.

• If 2 lts are supplementary to the same lt (or to
  lts), then they are .
• If lt1 lt2 are suppl lt2 lt3 are suppl, then lt1
  lt3.

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Proof of congruent suppl. Thm.

• Statements
• lt1 lt2 are suppl. lt2 lt3 are suppl.
• mlt1mlt2180o mlt2mlt3180o
• mlt1mlt2mlt2mlt3
• mlt1mlt3
• lt1 lt3

• Reasons
• Given
• defn,. of suppl lts
• Subst. prop of
• - prop of
• Defn. of lts

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Thm. 2.5 Congruent complements thm

• If 2 lts are complementary to the same lt (or to
  lts), then they are .
• If lt1 lt2 are complementary, and lt2 lt3 are
  complementary, then lt1 lt3.
• Proof is almost identical to the last thm., just
  change suppl. to compl.

1 2
3
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Postulate 12 Linear Pair Post.

• If 2 lts form a linear pair, then they are
  supplementary.
• lt1 lt2 are a linear pair, therefore they are
  suppl.

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Thm. 2.6 Vertical lts Thm.

• Vertical angles are .
• lt1 lt3 lt2 lt4.

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1 3
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Proof of Vertical lts thm

• Statements
• lt1 lt3 are vert.
• lt2 lt4 are vert.
• 2. lt1 lt2 are a lin pr. lt3 lt2 are a lin. pr.
• 3. lt1 lt2 are suppl lt3 lt2 are suppl.
• 4. lt1 lt3

• Reasons
• Given
• Defn. of Lin. Pr.
• Ln. Pr. Post.
• Suppls. Thm.

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Assignment