Lesson 5

More Postulates and Theorems About Lines and Planes

Definition of Parallel Lines

Coplanar lines that do not intersect are called parallel lines.

Definition of Perpendicular Lines

Perpendicular lines are lines that intersect to form a right angle.

This is the same definition for segments and rays.

Definition of Skew Lines

Skew lines are noncoplanar lines.Commonly you will see “and do not intersect” added to this definition, but is that really needed?No, because if the lines were to intersect they would be coplanar by Theorem 4-3

Theorem 5-1

If two parallel planes are cut by a third plane, then the lines of the intersection are parallel.

If plane A and B are parallel, then x ║y.

Theorem 5-2

If two lines in a plane are perpendicular to the same line, then they are parallel to each other.

If and , then…

Theorem 5-2

If two lines in a plane are perpendicular to the same line, then they are parallel to each other.

If and ,then .

Theorem 5-3

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other one.

If and , then…

Theorem 5-3

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other one.

If and , then .

Theorem 5-4

If two lines are perpendicular, then they form congruent adjacent angles.

If v ⊥ w, then ∠1 ≅ ∠2.

Theorem 5-5

If two lines form congruent adjacent angles, then they are perpendicular.

If ∠3 ≅ ∠4, then a ⊥ u.

Theorem 5-6

All right angles are congruent.

What two definition can you think of to prove this theorem true?

Def. of Right AnglesDef. of Congruent Angles

∠A &∠B are right angles

Then by the Def. of Right Angles, m∠A = 90° and m∠B = 90 °

By the Def. of Congruent Angles, ∠A ≅ ∠B

Postulate 10: The Parallel Postulate

Through a point not on a line, there exist exactly one line through the point that is parallel to the line.

Therefore, line f is the only line through point D that is parallel to line v.

Theorem 5-7: Transitive Property of Parallel Lines

If two lines are parallel to the same line, then they are parallel to each other.

If c║d & d║i, then c║ i.

Application in ConstructionMarty wants to hang ceiling tile in his game room. He wants each row of his tracks to be parallel to the first row. He measures the distance between the 1st & 2nd rows to verify they are parallel. However, his measure tape is not long enough to reach across the room. How can he ensure all rows are parallel to the 1st row?

Using Transitive Property of Parallel Lines he needs to make each row parallel to the previous row.What measurements did Marty take to verify the 1st & 2nd rows were parallel?

Questions/Review

Be sure not to assume that lines are parallel or perpendicular by just their appearance. Make sure to use symbols, postulates, theorems and/or definitions to justify your reasoning.

Be sure to refer to the diagram when one is given before you answer the question. If a diagram is not given, then try drawing one yourself.