Download Page 1 Name: Date: ______ 424 Class work Definition: Parallel

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Transcript
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.
pq
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.
pq
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, pq
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
pq
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.