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Parallel and Perpendicular Lines

Using Properties of Parallel Lines

Concept

Parallel Lines

Two lines that never intersect are said to be parallel. In the figure below, the lines and are parallel as are and

The notation for parallel lines is Thus, and can be written. Additionally, to show that lines are parallel, triangle hatch marks are drawn in a figure.

Rule

Parallel Postulate

The parallel postulate states that for a point, not on the line, there exists exactly one line through that is parallel to

Construction

Drawing a Line Parallel to a given Line and through a given Point

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The Parallel Postulate states that there exists exactly one line through a point, that is parallel to a line, This line can be found using a compass and a straightedge.

First, draw a line that intersects and goes through the point. The point where the new line intersects will be named

Next, fix the sharp end of the compass at and drawn an arc on between and With the same setting, draw an arc from across

Adjust the compass so that both ends align with the points where the first arc intersects both lines.

Using the setting above, place the sharp end of the compass where the second arc intersects Draw an arc intersecting the second arc. The point where these arcs intersect can be named

Draw the line that connects points and

This line is parallel to

info
Concept

Transversal

A transversal is a line that intersects two or more lines at different points.

Rule

Angles formed by a Transversal

When a transversal intersects two other lines, eight angles are created. These angles relate in different ways depending on their relative positions.

Vertical Angles

Corresponding Angles

Alternate Interior Angles

Alternate Exterior Angles

Rule

Corresponding Angles Theorem

If parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

In the diagram below,
This theorem can be proven using supplementary angles.

Proof

Corresponding Angles Theorem


To prove that corresponding angles are congruent, it will be shown that and are congruent.
Notice that, by definition, and are supplementary angles, as are and Thus, the following equations can be written. By transitivity, the equations can be set equal to each other. It follows then that Thus, The same reasoning applies to all pairs of corresponding angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of corresponding angles are congruent.
This can be summarized in the following flowchart proof.

Rule

Alternate Interior Angles Theorem

If parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

In the diagram below,
This theorem can be proven using vertical angles and corresponding angles.

Proof

Alternate Interior Angles Theorem


To prove that alternate interior angles are congruent, it will be shown that and are congruent.
Notice that, by definition, and are vertical angles. Thus, Notice also that, by definition, and are corresponding angles. Thus, By transitivity, Thus, it follows that The same reasoning applies to the other pair of alternate interior angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate interior angles are congruent.
This can be summarized in the following two-column proof.

Statement Reason
Vertical Angles Theorem
Corresponding Angles Theorem
Transitive Property of Equality
Rule

Alternate Exterior Angles Theorem

If parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

In the diagram below,
This theorem can be proven using corresponding angles and vertical angles.

Proof

Alternate Exterior Angles Theorem


To prove that alternate exterior angles are congruent, it will be shown that and are congruent.
Notice that, by definition, and are corresponding angles. Thus, Notice also that, by definition, and are vertical angles. Thus, By transitivity, Thus, it follows that The same reasoning applies to the other pair of alternate exterior angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate exterior angles are congruent.
This can be summarized in the following two-column proof.

Statement Reason
Corresponding Angles Theorem
Vertical Angles Theorem
Transitive Property of Equality
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Exercise

Determine the measures of and

Show Solution
Solution

To begin, notice that the diagram shows three parallel lines cut by a transversal. Thus, we can use the relevant angle relationships to determine the angle measures. Specifically,

We can use the given angle to determine the others. Let's name this

Let's begin with Notice that, by definition, and are corresponding angles. Thus, Next, notice that and are supplementary because they form a straight angle. Therefore, the sum of their measures is
Thus, Lastly, and are alternate interior angles. Thus, The angles marked in the figure have the following measures.
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