Geometry (A): READ: Perpendicular Lines Reading

Perpendicular Lines

Learning Objectives

Identify congruent linear pairs of angles
Identify the angles formed by perpendicular intersecting lines
Identify complementary adjacent angles
Identify the implications of perpendicular transversals on parallel lines.
Identify the converse theorems involving perpendicular transversals and parallel lines.
Understand and use the distance between parallel lines.

Introduction

Where they intersect, perpendicular lines form right $(90^\circ)$ angles. This lesson explores the different properties of perpendicular lines and how to understand them in various geometrical contexts.

Congruent linear pairs

A Linear Pair of Angles is a pair of adjacent angles whose outer sides form a straight line. The Linear Pair Postulate states that the angles of a linear pair are supplementary, that is, their measures must sum to $180^\circ.$ This makes sense because $180^\circ$ is the measure of a straight angle. When two angles that form a linear pair are congruent, there is only one possible measure for each of them— $90^\circ.$ Remembering that they must sum to $180^\circ,$ you can imagine how to find two equal angles. The easiest way to do this is to divide $180^\circ$ by $2$ , the number of congruent angles.

$180 \div 2 = 90$

Congruent angles that form a linear pair must each measure $90^\circ.$ You can use this information to fill in missing measures in diagrams and solve problems.

Example 1

What is the measure of $\angle {KLM}$ below?

Since the two angles form a linear pair, they must sum to $180^\circ.$ You can see that $\angle {MLN}$ is a right angle by the square marking in the angle, so that angle measures $90^\circ.$ Use subtraction to find the missing angle.

$m \angle {KLM} & = 180 - 90\\ m \angle {KLM} & = 90^\circ$

Angle $KLM$ will also equal $90^\circ$ because they are congruent linear angles.

Example 2

What is the measure of $\angle {LHK}$ below?

For now you can assume that the interior angles in a triangle must sum to $180^\circ$ (this is a fact that you have used in the past and we will prove it soon!).

Since you may assume that the interior angles in a triangle must sum to $180^\circ,$ you can find $m \angle {LHK}$ if you know the measures of the other two angles. Use the exterior right angle to find the measure of the interior angle adjacent to it. The two angles together are a linear pair. Since you know that the outer angle measures $90^\circ,$ find the value of the internal angle using subtraction.

$180 - 90 = 90$

$m \angle {LKH}$ will also equal $90^\circ$ because they are congruent linear angles. Now you know two of the internal angle measures in the triangle— $40^\circ$ and $90^\circ.$ Use subtraction to find the measure of $\angle {LHK}$ .

$m\angle{LHK} + 40 +90 &= 180 \\ m\angle{LHK} + 130 &=180 \\ m\angle{LHK} + 130 - 130 &= 180-130 \\ m\angle{LHK} &= 50\\ m\angle {LHK} & =50^\circ$

Intersecting Perpendicular Lines

We can extend what we just said about linear pairs to all pairs of congruent supplementary angles: Congruent supplementary angles will always measure $90^\circ$ each. When you have perpendicular lines, however, four different angles are formed.

Think about what you just learned. If two angles are a linear pair, and one of them measures $90^\circ,$ the other will also measure $90^\circ.$ Fill in the measures of $\angle{1}$ and $\angle{2}$ in the diagram to show that they are both right angles.

Now think back to what you learned earlier in this chapter. Vertical angles are two angles on opposite sides of intersecting lines. Applying the Vertical Angles Theorem, we know that vertical angles are also congruent. Using this logic you can prove that all four of the angles in this diagram are right angles.

Now that you know how to identify right angles formed by perpendicular lines, you can use this theorem for many different applications. Always be on the lookout for angles whose measures you know because of perpendicular lines.

Example 3

What is $m\angle{O}$ below?

Again you may assume that the interior angles in a triangle must sum to $180^\circ$

Since you know that the interior angles in a triangle must sum to $180^\circ,$ you can find $m \angle {O}$ if you know the measures of the other two angles. Use the exterior right angle to find the measure of its interior angle. Since the intersecting lines form one right angle, all angles formed will measure $90^\circ,$ and in particular, $m \angle {WHO} = 90^\circ$ . Now you know two of the internal angle measures in the triangle: $28^\circ$ and $90^\circ.$ Use subtraction to find $m\angle{O}$ .

$m\angle{O}+28+90&=180 \\ m\angle{O}+118&=180\\ m\angle{O}+118-118&=180-118\\ m\angle{O}&=62$

$\angle{O}$ measures $62^\circ$

Adjacent Complementary Angles

Remember that complementary angles are angles that sum to $90^\circ.$ If complementary angles are adjacent, they form perpendicular rays. You can then apply everything you have learned about perpendicular lines to the situation to find missing angle values.

Example 4

What is the measure of $\angle {MLK}$ in the diagram below?

$35^\circ+55^\circ=90^\circ$ , so the two angles in the upper right are complementary and sum to $90^\circ.$ Since $m \angle {OLN}$ is $90^\circ,$ its vertical angle will also measure $90^\circ.$ Therefore $m \angle {MLK} = 90^\circ$ .

Example 5

What is the measure of $\angle {TQU}$ in the diagram below?

Not to scale

Because the diagram is not to scale, you cannot find the measure just by looking at this diagram. $\angle {PQR}$ and $\angle {RQS}$ measure $70^\circ$ and $20^\circ$ respectively. Add these two angles to find their sum.

$70+20=90$

The two angles are complementary. Since $\angle {TQU}$ is vertical to the right angle, it must also be a right angle. So $m \angle {TQU}=90^\circ$ .

These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply concepts perpendicular lines to mathematical situations.

Points to Consider

Now that you understand perpendicular lines, you are going to take a closer look at perpendicular transversals.

Perpendicular Transversals

In the last lesson, you learned about perpendicular intersections. You know that when two lines are perpendicular they form four right $(90^\circ)$ angles. This lesson combines your knowledge of perpendicular lines with your knowledge of parallel lines and transversals.

Perpendicular Transversals and Parallel lines

When two lines are cut by a transversal, a number of special angles are formed. In previous lessons, you learned to identify corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. You learned that the lines crossed by the transversal are parallel if and only if corresponding angles are congruent. Likewise alternate interior and alternate exterior angles are congruent and interior angles on the same side of the transversal are supplementary if the lines crossed by the transversal are parallel. When the transversal is perpendicular, something interesting happens with these angles. Observe the angle measures in the example below.

Example 1

Lines $\overleftrightarrow{OR}$ and $\overleftrightarrow{KN}$ are parallel and $\overleftrightarrow{QT} \perp \overleftrightarrow{OR}.$

What is $m \angle {TSN}?$

Since you know that line $\overleftrightarrow{QT}$ is perpendicular to line $\overleftrightarrow{OR}$ , you can fill in the four right angles at that intersection.

The angle that corresponds to $\angle {TSN}$ is a right angle. This is true because you know lines $\overleftrightarrow{OR}$ and $\overleftrightarrow{KN}$ are parallel. Thus, the corresponding angles must be congruent. So $\angle {TSN}$ is a right angle. It measures $90^\circ.$

Notice in this example that if $\angle {QPO}$ is a right angle, then all of the angles formed by the intersection of lines $\overleftrightarrow{OR}$ and $\overleftrightarrow{QT}$ are right angles. Lines $\overleftrightarrow{KN}$ and $\overleftrightarrow{QT}$ are perpendicular as well. This is a result of the Corresponding Angles Postulate.

As in previous problems involving parallel lines crossed by a transversal, all pairs of angles remain either congruent or supplementary. When dealing with perpendicular lines, however, all of the angles are right angles.

Converse Theorem with Perpendicular Transversals

When examining the scenario of a perpendicular transversal with parallel lines, a converse theorem can be applied. The converse statement says that if a transversal forms right angles on two different coplanar lines, those two lines are parallel. Think back to the converse theorems you studied earlier in this chapter. They stated that if corresponding angles were congruent, consecutive interior angles were supplementary, or other specific relationship, then the two lines were parallel. Use this converse theorem to understand different graphic situations.

Example 2

Line $l$ below is a transversal, cutting through lines $k$ and $j$ .

Note: Figure NOT to scale

Are lines $j$ and $k$ parallel?

First, notice that the diagram is labeled “not to scale.” Do not make your decision based on how this diagram looks. Remember that if one angle at an intersection measures $90^\circ,$ all four angles measure $90^\circ.$ Fill in the angle measures you can identify with this information.

Since the corresponding angles are all $90^\circ,$ these two lines are parallel. The transversal is perpendicular to both lines $j$ and $k$ , so they must be parallel.

Distance Between Parallel Lines

When we talk about the distance between two points, what we are really talking about is the shortest or most direct distance between those two points. On paper, you can use a ruler or a taut string to find the distance between two points. When you measure the distance between a line and a point, the most direct path from a point to a line is always measured along the perpendicular from the point to the line.

Similarly, sometimes you might be asked to find the distance between two parallel lines. When you need to find this value, you need to find the length of a perpendicular segment that connects the two lines. Remember that if a line is perpendicular to one parallel line, it is perpendicular to both of them. Let’s look at an example on a coordinate grid.

Example 3

What is the distance between the two lines shown on the grid below?

In this image, the distance is found by drawing a perpendicular segment on the graph and calculating its length.

Because this line segment rises $4\;\mathrm{units}$ and has no run, it is $4\;\mathrm{units}$ long. The distance between the two lines on the graph is $4\;\mathrm{units.}$

Problems involving distances between lines will not always be this straightforward. You may have to use other skills and tools to solve the problem. The following example shows how you can apply tools from algebra to find the distance between two “slanted” parallel lines.

Example 4

What is the distance between the lines shown on the graph?

To start, you may think that you can just find the distance on the $y-$ axis between the two lines. That distance is $5\;\mathrm{units.}$ However, this is not correct, as the shortest distance between two lines will be a perpendicular segment between them. The $y-$ axis is perpendicular to neither line.

You’ll need to draw a perpendicular segment and calculate its length. To begin, identify the slope of the lines in the diagram. You can then identify the slope of a perpendicular because the slopes will be opposite reciprocals. To find the slope of a line, use the formula to calculate. Pick two points on one of the lines—here we will use $(0,4)$ and $(2,8)$ from the line $y = 2x + 4$ .

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(8 - 4)} {(2 - 0)}\\ \text{slope} & = \frac{4} {2}\\ \text{slope} & = 2$

The slope of the parallel lines in the diagram is $2.$ The slope of the perpendicular segment will be the opposite reciprocal of $2.$ The reciprocal of $2$ is $\frac{1} {2}$ and the opposite of $\frac{1} {2}$ is $-\frac{1} {2}$ . The slope of the perpendicular line is $-\frac{1} {2}$ . Pick a point on the line and use the slope to draw a perpendicular segment. Remember that the line will go down $1\;\mathrm{unit}$ for every $2\;\mathrm{units}$ you move the right.

Notice that there are points where the perpendicular segment intersects both parallel lines. It intersects the top line at $(0,4)$ and the bottom line at $(2,3).$ You need to find the length of this segment, and you know two points. You can use the distance formula which you learned in algebra and which we briefly reviewed in Chapter 1. Substitute the $x-$ and $y-$ coordinates from these points into the formula, and you’ll have the distance between the two parallel lines.

$d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ d &= \sqrt{(2 - 0)^2 + (3 - 4)^2}\\ d &= \sqrt{(2)^2 + (-1)^2}\\ d &= \sqrt{4 + 1}\\ d &= \sqrt{5}$

The distance between the two lines is $\sqrt{5}$ , or about $2.24\;\mathrm{units.}$

Lesson Summary

In this lesson, we explored perpendicular transversals. Specifically, we have learned:

The properties of congruent angles that form a linear pair.
How to identify the angles formed by perpendicular intersecting lines.
How to identify complementary adjacent angles.

How to identify the implications of perpendicular transversals on parallel lines.
How to identify the converse theorems involving perpendicular transversals and parallel lines.
To understand and use the distance between parallel lines.

These will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply the concepts of perpendicular transversals to new mathematical situations.

Points to Consider

Finding the distance from a point to a line and the distance between two lines are two good examples of ways to apply skills you learned in algebra, such as finding slopes and using the distance formula to geometry problems. Even when geometric problems are given without a coordinate system, you can often define a convenient coordinate system to help you solve the problem.

Last modified: Monday, June 28, 2010, 1:40 PM