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3. Parallel & Perpendicular Lines
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Geometry

3.03 Perpendicular lines

Lesson
Practice

Proofs and constructions of perpendicular lines

Perpendicular lines

Two lines that intersect at right angles. Lines are denoted as being perpendicular by the symbol \perp.

Two intersecting lines with a right angle marking.
Midpoint

A point exactly halfway between the endpoints of a segment that divides it into two congruent line segments.

A line segment with endpoints A and C with a point B directly in the middle labeled Midpoint.

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

  1. Choose two points on a given line.

    A line with two points.

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

    A line with two points. An arc centered at one of the points is drawn.

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

    A line with two points. Two arcs centered at the points are drawn.

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

    A line with two points. Two arcs centered at the points are drawn. The points of intersections of the two arcs are labeled A and A prime.

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

    A line with two points. Two arcs centered at the points are drawn. The points of intersections of the two arcs are labeled A and A prime. A line passing through A and A prime is drawn.

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

Examples

Example 1

Construct a proof of the following:

Given m \angle1 = m \angle 2

Prove: l \perp m

Line l intersecting line m. Line m forms an angle labeled 1 with the left side of line l, and an angle labeled 2 with the right side of line l.
Worked Solution
Create a strategy

We can use a flow chart proof, two column proof, bulleted list, or paragraph proof to prove that l \perp m. For this proof, we will use a two column proof.

Apply the idea
To prove: l \perp m
StatementsReasons
1.m \angle1 = m \angle 2Given
2.\angle 1 and \angle 2 are supplementaryLinear pair postulate
3.m\angle1 + m\angle 2 = 180 \degreeDefinition of supplementary angles
4.m\angle1 + m\angle 1 = 180 \degreeSubstitution property of equality
5.m\angle1= 90 \degreeDivision property of equality
6.\angle 1 is a right angle and l \perp mDefinition of perpendicular lines

Example 2

Ludek is creating a shirt design on his computer. He wants to include some perpendicular lines as part of the design, but the program he is using doesn't measure angles so he can't construct them directly using a right angle.

The program Ludek is using can construct lines and circles accurately through or centered at points. Describe how Ludek could construct the perpendicular lines for his design.

Worked Solution
Create a strategy

One approach Ludek can use to construct perpendicular lines is by using dynamic technology software.

Apply the idea

  1. Use the 'Line' tool to construct a line through two points.

    A screenshot of the GeoGebra geometry tool showing the constructions described in step 1. Speak to your teacher for more details.

  2. Use the 'Circle with Center' tool to construct a circle about each point on the line, such that the circles intersect at two distinct points.

    A screenshot of the GeoGebra geometry tool showing the constructions described in step 2. Speak to your teacher for more details.

  3. Use the 'Line' tool once more to construct a line through the points of intersection of the circles. This line will be perpendicular to the first line.

    A screenshot of the GeoGebra geometry tool showing the constructions described in step 3. Speak to your teacher for more details.
Idea summary

We can use the relationships between angles to prove that lines are perpendicular. We can use various methods to construct perpendicular lines including technology and a compass and straightedge.

Solve problems with perpendicular lines

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 (because both angles in any pair of corresponding angles will measure 90 \degree making them congruent).

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

Perpendicular tranversal 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.

Examples

Example 3

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.
Worked Solution
Create a strategy

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 \left(x+38\right)\degree needs to be a right angle.

Apply the idea

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 4

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.

Worked Solution
Apply the idea

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

Reflect and check

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.

Worked Solution
Apply the idea

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.

Reflect and check

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

Idea summary

The perpendicular transversal theorem and its converse help us draw conclusions about relationships between lines and solve problems.

  • The perpendicular transversal theorem states that if a \perp c and a \parallel b, then b \perp c.
  • The converse of perpendicular transversal theorem states that if a \perp c and b \perp c, then a \parallel b.

Outcomes

G.RLT.2

The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal.

G.RLT.2c

Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal.

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