4.6 Proofs Involving Parallel Lines DEFINITIONS/THEOREMS COMMONLY USED IN PARALLEL LINE PROOFS Condition Present Definitions/Theorems to Be Written in the "Reasons" Column of Proofs (write in words) Conclusion Drawn from the Definition/Theorem to Be Written in the "Statements" Column of Proofs (use symbols) 1 Given Parallel Lines cut by a Transversal Theorem: Parallel lines cut by a transversal form congruent alternate interior angles. State the congruent alternate interior angles. Theorem: Parallel lines cut by a transversal form congruent alternate exterior angles. State the congruent alternate exterior angles. Theorem: Parallel lines cut by a transversal form congruent corresponding angles. State the congruent corresponding angles. Theorem: Parallel lines cut by a transversal form supplementary same side interior angles. State the supplementary same side interior angles. Note: When given parallel lines, our theorems start with the word "parallel." 2 Proving Lines cut by a Transversal are Parallel Theorem: If two lines cut by a transversal form congruent alternate interior angles, then the lines are parallel. State which lines are //. Theorem: If two lines cut by a transversal form congruent alternate exterior angles, then the lines are parallel. Note: When proving lines are parallel, our theorems end with the word "parallel." Theorem: If two lines cut by a transversal form congruent corresponding angles, then the lines are parallel. Note: The theorems used when proving lines parallel are converses of the theorems used when we're given parallel lines. Ex: The converse of "parallel lines -> alternate interior angles" is "alternate interior angles -> parallel lines". Theorem: If two lines cut by a transversal form supplementary same side interior angles, then the lines are parallel. Експортувати в Таблиці Parallel Lines Congruent Alternate Interior Angles Congruent Alternate Exterior Angles Congruent Corresponding Angles Supplementary Same Side Interior Angles Use the given diagram where AB || CD to identify each pair of the angles described to the right: congruent alternate interior angles: angle ____ and angle ____; angle ____ and angle ____ congruent alternate exterior angles: angle ____ and angle ____; angle ____ and angle ____ congruent corresponding angles: angle ____ and angle ____; angle ____ and angle ____; angle ____ and angle ____; angle ____ and angle ____ supplementary same side interior angles: angle ____ and angle ____; angle ____ and angle ____ Euclidean Geometry Proofs involving Parallel Lines 1.) Given: AFCD, AB is perpendicular to BC, DE is perpendicular to EF, BC || EF, AB = DE Prove: AC = FD Statements Reasons 1. AFCD 1. Given 2. AB is perpendicular to BC, DE is perpendicular to EF 2. Given 3. angle B and angle E are right angles 3. 4. angle B = angle E 4. All right angles are congruent. 5. BC EF 6. angle BCA = angle EFD 6. 7. AB = DE 7. Given 8. triangle ABC = triangle DEF 8. 9. AC = FD 9.