Using Properties of Parallel Lines

Concept

Parallel Lines

Two lines that never intersect are said to be parallel. In the figure below, the lines $A B$ and $C D$ are parallel as are $E F$ and $G H .$

The notation for parallel lines is $∥ .$ Thus, $A B ∥ C D$ and $E F ∥ G H$ 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, $P,$ not on the line, $l,$ there exists exactly one line through $P$ that is parallel to $l .$

Construction

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

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

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

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

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 $Q P .$ Draw an arc intersecting the second arc. The point where these arcs intersect can be named $R .$

Draw the line that connects points $P$ and $R .$

This line is parallel to $l .$

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: angles that lie on opposite sides of a point of intersection.
corresponding angles: angles formed by different points of intersection that lie in the same position.
alternate interior angles: angles that lie inside the parallel lines and on opposite sides of the transversal.
alternate exterior angles: angles that lie outside the parallel lines and on opposite sides of the transversal.

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,

∠ 1 ≅ ∠ 5, ∠ 2 ≅ ∠ 6, ∠ 3 ≅ ∠ 7, and ∠ 4 ≅ ∠ 8 .

This theorem can be proven using supplementary angles.

Proof

Corresponding Angles Theorem

To prove that corresponding angles are congruent, it will be shown that $∠ 1$ and $∠ 5$ are congruent.
Notice that, by definition, $∠ 1$ and $∠ 4$ are supplementary angles, as are $∠ 4$ and $∠ 5 .$ Thus, the following equations can be written. $m ∠ 1 + m ∠ 4 = 18 0^{\circ} m ∠ 4 + m ∠ 5 = 18 0^{\circ}$ By transitivity, the equations can be set equal to each other. $m ∠ 1 + m ∠ 4 = m ∠ 4 + m ∠ 5$ It follows then that $m ∠ 1 = m ∠ 5 .$ Thus, $∠ 1 ≅ ∠ 5 .$ 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,

∠ 3 ≅ ∠ 5, and ∠ 4 ≅ ∠ 6 .

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 $∠ 3$ and $∠ 5$ are congruent.
Notice that, by definition, $∠ 1$ and $∠ 3$ are vertical angles. Thus, $m ∠ 1 = m ∠ 3 .$ Notice also that, by definition, $∠ 1$ and $∠ 5$ are corresponding angles. Thus, $m ∠ 1 = m ∠ 5 .$ By transitivity, $m ∠ 3 = m ∠ 5 .$ Thus, it follows that $∠ 3 ≅ ∠ 5 .$ 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
$∠ 1 ≅ ∠ 3$	Vertical Angles Theorem
$∠ 1 ≅ ∠ 5$	Corresponding Angles Theorem
$∠ 3 ≅ ∠ 5$	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,

∠ 1 ≅ ∠ 7, and ∠ 2 ≅ ∠ 8 .

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 $∠ 1$ and $∠ 7$ are congruent.
Notice that, by definition, $∠ 1$ and $∠ 5$ are corresponding angles. Thus, $m ∠ 1 = m ∠ 5 .$ Notice also that, by definition, $∠ 5$ and $∠ 7$ are vertical angles. Thus, $m ∠ 5 = m ∠ 7 .$ By transitivity, $m ∠ 1 = m ∠ 7 .$ Thus, it follows that $∠ 1 ≅ ∠ 7 .$ 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
$∠ 1 ≅ ∠ 5$	Corresponding Angles Theorem
$∠ 5 ≅ ∠ 7$	Vertical Angles Theorem
$∠ 1 ≅ ∠ 7$	Transitive Property of Equality

Determine the angle measures

fullscreen

Exercise

Determine the measures of $∠ A,$ $∠ B,$ and $∠ C .$

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,

corresponding angles are congruent, and
alternate interior angles are congruent.

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

Let's begin with

∠ A .

Notice that, by definition,

∠ D

and

∠ A

are corresponding angles. Thus,

m ∠ A = m ∠ D = 6 3^{\circ} .

Next, notice that

∠ D

and

∠ B

are supplementary because they form a straight angle. Therefore, the sum of their measures is

18 0^{\circ} .

m ∠ D + m ∠ B = 18 0^{\circ}

Substitute

m ∠ D = 6 3^{\circ}

6 3^{\circ} + m ∠ B = 18 0^{\circ}

SubEqn

LHS - 6 3^{\circ} = RHS - 6 3^{\circ}

m ∠ B = 11 7^{\circ}

Thus,

m ∠ B = 11 7^{\circ} .

Lastly,

∠ D

and

∠ C

are alternate interior angles. Thus,

∠ C ≅ ∠ D .

m ∠ C = m ∠ D = 6 3^{\circ}

The angles marked in the figure have the following measures.

m ∠ A = 6 3^{\circ} m ∠ B = 11 7^{\circ} m ∠ C = 6 3^{\circ}

Construction

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

Proof

Proof

Proof

Example

Determine the angle measures