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Name: __________________ 424 Class work Definition: Parallel lines are coplanar lines that do not intersect. Skew lines are non coplanar lines, therefore they do not intersect. Parallel planes are planes that do not intersect. Date: _________ From the diagram above we can also conclude that a line and a plane are parallel if they do not intersect. Theorem: If two parallel planes are cut by a third plane, then the lines of intersection are parallel. Given: Plane A ∥ plane B; Plane C intersects A in line MN ; Plane C intersects B in line RS . Prove: MN RS Definition: A transversal is a line that intersects two or more coplanar lines in different points. Alternate interior angles are two nonadjacent interior angles on opposite sides of the transversal. Same-side interior angles are two interior angles on the same side of the transversal. Corresponding angles are two angles in corresponding positions relative to the two lines. Postulate In a plane, if two lines are parallel and cut by a transversal, then the corresponding angles are congruent. If two lines are coplanar and cut by a transversal such that the corresponding angles are congruent, then the two lines are parallel. Theorem: In a plane, if two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. Proof: Given: Transversal t cuts p and q, p q Prove: ∠1 ≅ ∠2 Statement 1. Transversal t cuts p and q, p q 2. ∠2 ≅ ∠3 Reason 1. Given 2. If two lines are coplanar and cut by a transversal such that the corresponding angles are congruent, then the two lines are parallel. 3. ∠3 ≅ ∠1 3. Vertical angle theorem 4. ∠1 ≅ ∠2 4. Transitive axiom Theorem: In a plane, if two lines are cut by a transversal such that the alternate interior angles are congruent, then the two lines are parallel. Proof: Given: Transversal t cuts p and q, ∠1 ≅ ∠2 Prove: p q Statement Reason 1. Transversal t cuts p and q, ∠1 ≅ ∠2 1. Given 2. ∠2 ≅ ∠3 2. Vertical angle theorem 3. ∠1 ≅ ∠3 3. Transitive axiom 4. pq 4. If two lines are coplanar and cut by a transversal such that the corresponding angles are congruent, then the two lines are parallel. Theorem: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Proof: Given: p q transversal t cuts p and q Prove: ∠1 is supplementary to ∠4 Statement p q transversal t cuts p and q 2. ∠1 ≅ ∠2 1. m∠1 = m∠2 ∠2 is supplementary to ∠4 m∠2 + m∠4 = 180 6. m∠1 + m∠4 =180 7. ∠1 is supplementary to ∠4 3. 4. 5. Reason 1. Given 2. If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. 3. Definition of congruency 4. Supplement Axiom 5. Definition of supplementary angles. 6. Substitution 7. Definition of supplementary angles. Theorem: In a plane, if two lines are cut by a transversal such that the interior angles on the same side of the transversal are supplementary, then the two lines are parallel. Proof: Given: Transversal t cuts p and q ∠1 is supplementary to ∠2 Prove: p q Statement Reason 1. Transversal t cuts p and q 1. Given 2. ∠1 is supplementary to ∠2 2. Given 3. ∠3 is supplementary to ∠2 3. Supplement Axiom 4. ∠1 ≅ ∠3 5. pq 4. If two angles are supplementary to the same angle then they are congruent. 5. In a plane, if two lines are cut by a transversal such that the alternate interior angles are congruent, then the two lines are parallel. Theorem: In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Proof: Given: Transversal t cuts p and q t ⊥ p, pq Prove: t ⊥ q Statement 1. Transversal t cuts p and q, t ⊥ p , p q 2. 3. 4. ∠1 Is a right angle m∠1 =90 ∠1 ≅ ∠2 5. 6. 7. 8. m∠1 = m∠2 m∠2 = 90 ∠2 is a right angle t⊥q Reason 1. Given 2. Definition of perpendicular lines. 3. Definition of a right angle. 4. If two lines are parallel and cut by a transversal, then the corresponding angles are congruent. 5. Definition of congruency 6. Substitution 7. Definition of a right angle. 8. Definition of perpendicular lines. Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel. Proof: Given: Transversal t cuts p and q t ⊥ p, t ⊥q Prove: p q Statement 1. Transversal t cuts p and q, t ⊥ p , t ⊥ q 2. 3. 4. ∠1 and ∠2 are right angles ∠1 ≅ ∠2 pq Reason 1. Given 2. Definition of perpendicular lines. 3. All right angles are congruent. 4. If two lines are coplanar and cut by a transversal such that the corresponding angles are congruent, then the two lines are parallel. Theorem: if two lines are parallel to the same line, then they are parallel to each other. Proof: Given: n m; m l Prove: n l Statement 1. n m; m l 2. ∠1 ≅ ∠2; ∠2 ≅ ∠3 3. ∠1 ≅ ∠3 4. n l Reason 1. Given 2. If two parallel lines are cut by a transversal, then their corresponding angles are congruent. 3. Transitive property. 4. In a plane, if two lines are cut by a transversal such that the corresponding angles are congruent, then the lines are parallel.