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2. Parallel & Perpendicular Lines
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United States of AmericaFL
Grade 912

2.04 Parallel and perpendicular lines

Lesson
Practice
Lesson

Concept summary

A particular type of transversal is one that is perpendicular to the lines that it intersects.

A pair of lines intersected by a transversal. The two lines form right angles with the transversal.

Using the converse of corresponding angles postulate, we can determine that the two lines intersected by the transversal are parallel.

We can also draw conclusions about which pairs of lines are parallel or perpendicular in this diagram.

When the transveral is perpendicular to the lines it intersect, we can get the following theorems relating parallel and perpendicular lines.

Perpendicular transversal theorem

If a\perp c and a\parallel b, then b\perp c.

Parallel lines a and b intersecting line c. Lines a and b are perpendicular to c.
Converse of perpendicular transversal theorem

If a\perp c and b\perp c, then a\parallel b.

Parallel lines a and b intersecting line c. Lines a and b are perpendicular to c.

Using properties of parallel lines, we can relate three parallel lines given one line in common.

Transitivity of parallelism

If a\parallel b and b\parallel c, then a\parallel c.

Lines a, b, and c. Line a is perpendicular to line b. Line b is perpendicular to line c.

Worked examples

Example 1

Consider the given relations between lines:

  • a\parallel b
  • b\perp d
  • c\perp d
a

Determine the relationship between the lines b and c. Justify your answer.

Solution

Using the converse of perpendicular transversal theorem, since b\perp d and c\perp d, we have that b\parallel c.

Reflection

We can use the given theorems to justify, but we can also check our reasoning by sketching a diagram.

Parallel horizontal lines a and b, a vertical line d, and another line c. Line d forms right angles with both b and c.

By extending some of these lines and using other theorems to show parallelism and perpendicularity, we can reach the same results.

b

Determine the relationship between the lines a and d. Justify your answer.

Solution

Since parallel relations are symmetric, since a \parallel b we can write that b\parallel a.

Using the perpendicular transversal theorem, since b\perp d and b\parallel a, we have that d\perp a.

Reflection

Since perpendicular relations are also symmetric, we can also write the relation as a\perp d.

c

Determine the relation between the lines a and c. Justify your answer.

Solution

We are given that a\parallel b and determined in part (a) that b\parallel c.

Using the transitivity of parallelism, since a\parallel b and b\parallel c, we have that a\parallel c.

Example 2

Find the value of x that makes the diagram valid.

Line a intersects with line b, and line c intersects with line d. Lines a and c are parallel, as well as lines b and d. Lines a and b form a right angle. Lines c and d form an angle labeled x plus 38 degrees.

Approach

Using the perpendicular transversal theorem and its converse, we can conclude that the diagram is valid when a and c are perpendicular to b and d. This means that the angle of measure (x+38)\degree needs to be a right angle.

Solution

To find the value of x that makes the diagram valid, we let:

x+38=90

Solving this equation gives us x=52, which is when the diagram will be valid.

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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