Obj. 12 Proving Lines Parallel
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1. The document discusses using angles formed by a transversal to show that lines are parallel using the parallel line theorems. 2. It provides an example of finding values of x and y that make two sets of lines parallel using the same-side interior angle theorem. 3. It also shows a proof that lines CD and BE are parallel given that angle BE bisects angle DBA and angle 3 is congruent to angle 1 using alternate interior angles.
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The document discusses how to prove lines are parallel using angles formed by a transversal. It provides four theorems involving corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles that can be used to show lines are parallel if the angles are congruent or supplementary. An example problem demonstrates finding values of x and y that make two sets of lines parallel by using the consecutive interior angles theorem. The document also includes a multi-step proof involving bisecting an angle and using the alternate interior angles theorem.
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The document discusses how to prove that lines are parallel using angles formed by a transversal. It provides the following parallel postulates: corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary. An example problem demonstrates finding values of x and y that make two sets of lines parallel using these postulates. The document also discusses the use of algebraic properties in geometric proofs.
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Obj. 12 Proving Lines Parallel
- 1. Obj. 12 Proving Lines Parallel The student is able to (I can): • Use angles formed by a transversal to show that lines are parallel.
- 2. Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converses of the parallel line theorems can be used to prove lines parallel • Corresponding Angles • Alternate Interior Angles • Alternate Exterior Angles — If angles are congruent, then the lines are parallel. • Same-Side Interior Angles — If angles are supplementary, then the lines are parallel.
- 3. Example Find values of x and y that make the red lines parallel and the blue lines parallel. (x−40)° (x+40)° If the blue lines are parallel, then the same-side interior angles must be supplementary. x − 40 + x + 40 = 180 2x = 180 x = 90 y°
- 4. Example Find values of x and y that make the red lines parallel and the blue lines parallel. (x−40)° (x+40)° If the red lines are parallel, then the same-side interior angles must be supplementary. 90 − 40 + y = 180 50 + y = 180 y = 130 y°
- 5. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A Plan of proof: Because bisects ÐDBA, BE Ð2 @ Ð3. Because Ð3 @ Ð1, I can set Ð2 @Ð1 using substitution. Ð1 and Ð2 are alternate interior angles, and since they are congruent, the lines are parallel.
- 6. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. B E bisects ÐDBA 1. Given
- 7. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. BE bisects ÐDBA 1. Given 2. Ð2 @ Ð3 2. Def. Ð bisector
- 8. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. BE bisects ÐDBA 1. Given 2. Ð2 @ Ð3 2. Def. Ð bisector 3. Ð3 @ Ð1 3. Given
- 9. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. BE bisects ÐDBA 1. Given 2. Ð2 @ Ð3 2. Def. Ð bisector 3. Ð3 @ Ð1 3. Given 4. Ð2 @ Ð1 4. Subst. prop. @
- 10. Given: bisects ÐDBA; Ð3 @ Ð1 Prove: BE CD BE C B 2 3 1 D E A SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. BE bisects ÐDBA 1. Given 2. Ð2 @ Ð3 2. Def. Ð bisector 3. Ð3 @ Ð1 3. Given 4. Ð2 @ Ð1 4. Subst. prop. @ 5. 5. Conv. alt. int. Ðs CD BE