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Using Properties of Parallel Lines

Concept

Parallel Lines

Two lines that never intersect are said to be parallel. In the figure below, the lines AB\overleftrightarrow{AB} and CD\overleftrightarrow{CD} are parallel as are EF\overleftrightarrow{EF} and GH.\overleftrightarrow{GH}.

The notation for parallel lines is .\parallel\text{.} Thus, ABCD\overleftrightarrow{AB}\parallel \overleftrightarrow{CD} and EFGH\overleftrightarrow{EF}\parallel\overleftrightarrow{GH} 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,P, not on the line, l,l, there exists exactly one line through PP that is parallel to l.l.

Construction

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


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

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

Next, fix the sharp end of the compass at QQ and drawn an arc on QP\overleftrightarrow{QP} between QQ and P.P. With the same setting, draw an arc from PP across QP.\overleftrightarrow{QP}.

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 QP.\overleftrightarrow{QP}. Draw an arc intersecting the second arc. The point where these arcs intersect can be named R.R.

Draw the line that connects points PP and R.R.

This line is parallel to l.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

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, 15,26,37, and 48.\angle 1 \cong \angle 5, \angle 2 \cong \angle 6, \angle 3 \cong \angle 7, \text{ and } \angle 4 \cong \angle 8.
This theorem can be proven using supplementary angles.

Proof

info
Corresponding Angles Theorem


To prove that corresponding angles are congruent, it will be shown that 1\angle 1 and 5\angle 5 are congruent.
Notice that, by definition, 1\angle 1 and 4\angle 4 are supplementary angles, as are 4\angle 4 and 5.\angle 5. Thus, the following equations can be written. m1+m4=180m4+m5=180\begin{aligned} m\angle 1 + m\angle 4 = 180 ^\circ \\ m\angle 4 + m\angle 5 = 180 ^\circ \end{aligned} By transitivity, the equations can be set equal to each other. m1+m4=m4+m5 m\angle 1 + m\angle 4 = m\angle 4 + m\angle 5 It follows then that m1=m5.m\angle 1 = m\angle 5. Thus, 15.\angle 1 \cong \angle 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, 35, and 46.\angle 3 \cong \angle 5, \text{ and }\angle 4 \cong \angle 6.
This theorem can be proven using vertical angles and corresponding angles.

Proof

info
Alternate Interior Angles Theorem


To prove that alternate interior angles are congruent, it will be shown that 3\angle 3 and 5\angle 5 are congruent.
Notice that, by definition, 1\angle 1 and 3\angle 3 are vertical angles. Thus, m1=m3. m\angle 1 = m\angle 3. Notice also that, by definition, 1\angle 1 and 5\angle 5 are corresponding angles. Thus, m1=m5. m\angle 1 = m\angle 5. By transitivity, m3=m5.m\angle 3 = m\angle 5. Thus, it follows that 35.\angle 3 \cong \angle 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
13\angle 1\cong \angle 3 Vertical Angles Theorem
15\angle 1 \cong \angle 5 Corresponding Angles Theorem
35\angle 3 \cong \angle 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, 17, and 28.\angle 1 \cong \angle 7, \text{ and }\angle 2 \cong \angle 8.
This theorem can be proven using corresponding angles and vertical angles.

Proof

info
Alternate Exterior Angles Theorem


To prove that alternate exterior angles are congruent, it will be shown that 1\angle 1 and 7\angle 7 are congruent.
Notice that, by definition, 1\angle 1 and 5\angle 5 are corresponding angles. Thus, m1=m5. m\angle 1 = m\angle 5. Notice also that, by definition, 5\angle 5 and 7\angle 7 are vertical angles. Thus, m5=m7. m\angle 5 = m\angle 7. By transitivity, m1=m7.m\angle 1 = m\angle 7. Thus, it follows that 17.\angle 1 \cong \angle 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
15\angle 1\cong \angle 5 Corresponding Angles Theorem
57\angle 5 \cong \angle 7 Vertical Angles Theorem
17\angle 1 \cong \angle 7 Transitive Property of Equality
Exercise

Determine the measures of A,\angle A, B,\angle B, and C.\angle C.

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 D.\angle D.

Let's begin with A.\angle A. Notice that, by definition, D\angle D and A\angle A are corresponding angles. Thus, mA=mD=63. m\angle A=m\angle D = 63^\circ. Next, notice that D\angle D and B\angle B are supplementary because they form a straight angle. Therefore, the sum of their measures is 180.180^\circ.
mD+mB=180m\angle D + m\angle B =180^\circ
63+mB=180{\color{#0000FF}{63^\circ}} + m\angle B = 180^\circ
mB=117m\angle B = 117^\circ
Thus, mB=117.m\angle B=117^\circ. Lastly, D\angle D and C\angle C are alternate interior angles. Thus, CD.\angle C \cong \angle D. mC=mD=63 m\angle C=m\angle D = 63^\circ The angles marked in the figure have the following measures. mA=63mB=117mC=63\begin{aligned} &m\angle A = 63^\circ \\ &m\angle B = 117^\circ \\ &m\angle C= 63^\circ \end{aligned}
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