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2. Parallel & Perpendicular Lines
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Geometry

2.04 Parallel and perpendicular lines

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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.

To construct a perpendicular line, we can use the following method:

  1. Choose two points on a given line.

  2. Construct an arc centered at one of the points.

  3. Construct an arc centered at the other point, such that this arc intersects the arc constructed in Step 2 at two distinct points.

  4. Plot points where the two arcs intersect. Here we have used A and A'.

  5. Construct the line \overleftrightarrow{AA'}. It is perpendicular to the line constructed in Step 1.

If we want the perpendicular line to pass through a particular point, we can adjust the radii of the compass so that both arcs pass through that point.

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.

Example 3

Urbana is looking for some treasure that her aunt has buried on the beach. Her aunt has given the hint that the treasure is 20 steps directly North of their bags. Urbana knows that when she is standing at the bags and facing towards the nearest yellow flag she is looking directly East.

Describe how Urbana can determine which direction is directly north.

Approach

The directions East and North are perpendicular, so Urbana wants to find the direction that is perpendicular to when she is facing the yellow flag.

Solution

Since Urbana knows when she looking directly East, she can draw a line in the sand that points East.

Urbana can draw two arcs centered at different points on the line. One way that Urbana can ensure that the arcs keep a consistent radius is to pin one end of a towel to the center point and draw an arc in the sand at the other end of the towel. The length of the towel will act as a consistent radius for the arc.

To find the line that points directly North from the bags, she must draw the arcs so that they pass through the location of the bags. She must also choose points on the line that are close enough for the arcs to intersect at two distinct points.

The line which passes through the two points of intersection of the arcs will run directly in the North-South directly. Using this line to guide her, Urbana can move directly North to find the treasure.

Reflection

When sketching out the scenario, you will notice that the East-facing line that Urbana constructs at the start should not pass through the location of the bags. This is because doing so makes it impossible to construct two arcs that both pass through the bags and also have two distinct points of intersection.

Outcomes

G.CO.C.8

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

G.CO.D.11

Perform formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

G.CO.D.12

Use geometric constructions to solve geometric problems in context, by hand and using technology.*

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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