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Theorems about Parallel Lines

Theorems about Parallel Lines. Objectives . You will be able to use theorems and definitions to find the measures of angles. You will be able to use theorems and definitions to write a formal proof. Vocabulary. Indirect Proof –

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Theorems about Parallel Lines

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  1. Theorems about Parallel Lines

  2. Objectives • You will be able to use theorems and definitions to find the measures of angles. • You will be able to use theorems and definitions to write a formal proof.

  3. Vocabulary • Indirect Proof – • A proof that shows that the conclusion cannot be false because accepted facts would be contradicted.

  4. Theorem – If two lines are parallel, then the interior angles on the same side of the transversal are supplementary. • Proof by contradiction (Indirect Proof) • To prove: 4 and 6 are supplementary

  5. Reason Given Indirect proof assumption (something taken to be true without proof), the opposite of “To Prove”. Def of angles that are not supplementary Definition of not equal, greater than or less than Euclid’s Postulate 5 Statement • n II m, transversal t • 4 and 6 are not supplementary • 4 + 6  180o • 4 + 6 < 180o or 4 + 6 > 180o • n must intersect m This contradicts the given n II m Theorem is valid and proof is complete

  6. Theorem: If two lines cut by a transversal are parallel, then the corresponding angles are equal. To Prove: 2 = 6 • n II m, transversal t • 4 + 6 = 180o • 2 + 4 = 180o • 4 + 6 = 2 + 4 •  2 = 6 Given Theorem, interior on the same side of the transversal are supplementary Adjacent angles whose exterior sides are a straight line are supplementary. Definition of adjacent and supplementary. Axiom 1, things equal to the same thing are equal to each other. Axiom 3, equals subtracted from equals are equal.

  7. Theorem: If two lines cut by a transversal are parallel then the alternate interior angles are equal. To Prove: 2 = 4 • n II m, transversal t • 6 is supplementary to 2; 2 + 6 = 180o • 4 + 6 = 180o • 2 + 6 = 4 + 6 •  2 = 4 Given Definition of adjacent & supplementary Theorem: interior angles on the same side of the transversal are supplementary Axiom 1, things equal to the same thing are equal to each other. Steps 2 & 3 Axiom 3, equals subtracted from equals are equal.

  8. Alternate Interior Angles Postulate • If a transversal intersects two lines so that the alternate interior angles are equal, then the lines are parallel.

  9. Practice • The m7 is three times that of 6. • 7 = 135o ;6 = 45o. • Eight times the m 5 = m8. • 5 = 20o ; 8 = 160o

  10. Supplementary Angle Practice Find the measure of the second angle: • One angle = 52o • 128o • One angle = 28o • 152o • One angle = 36o • 144o • One angle = 63o • 117o • One angle = 107o • 73o

  11. Find the measures of the angles • 1 = 125o • 2 = • 3 = • 4 = • 5 = • 6 = • 7 = • 8 =

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