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3. Parallel & Perpendicular Lines
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Geometry

3.01 Parallel lines and transversals

Lesson
Practice

Parallel lines and transversals

When one line intersects a pair of lines (or more), we refer to it as a transversal.

Transversal

A line that intersects two or more lines in the same plane at different points.

A pair of lines both intersected by a line that is the transversal.

When a transversal cuts through a pair of lines, it allows us to pair up and name the angles that are formed.

Corresponding angles

Angles that are in the same position on two lines in relation to a transversal.

A pair of lines intersected by a transversal. Two marked angles lie on the same side of the transversal, and on the same sides of the lines.
Same-side (consecutive) interior angles

Angles that are on the interior of two lines on the same side of the transversal.

A pair of lines intersected by a transversal. Two marked angles lie on the same side of the transversal, between the two lines.
Same-side (consecutive) exterior angles

Angles that are on the exterior of two lines on the same side of the transversal.

A pair of lines intersected by a transversal. Two marked angles lie on the same side of the transversal, outside the two lines.
Alternate interior angles

Angles that are on the interior of two lines on different lines and opposite sides of the transversal.

A pair of lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, between the two lines.
Alternate exterior angles

Angles that are on the exterior of two lines on different lines and opposite sides of the transversal.

A pair of lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, outside the two lines.

Exploration

Check the parallel lines box, then use the points to drag the transversal and the parallel lines.

Loading interactive...

  1. What relationship do each of the types of angle pairs have?

  2. Uncheck the parallel lines box and drag the lines. Are the relationships stated in the previous question still true? How do you know?

  3. Check the parallel lines box again and drag one of the parallel lines to create a translation. How can you use a translation to verify the relationships you found?

Parallel lines

Two lines that never intersect. Lines are denoted as being parallel by the symbol \parallel.

A pair of non intersecting lines.

When lines cut by the transversal are parallel, the angle pairs created have special relationships. They will either be congruent or supplementary.

Corresponding angles theorem

If a transversal intersects two parallel lines, then corresponding angles are congruent.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, and on the same sides of the parallel lines. The two angles are congruent.

We can use our knowledge of translations to show this theorem is true. We can imagine translating one of the angles along the traversal until it meets the second parallel line. It will match the corresponding angle exactly.

The corresponding angles theorem can be used as a basis for proving relationships between other angle pairs, as given in the following theorems.

Consecutive interior angles theorem

If a transversal intersects two parallel lines, then same-side (consecutive) interior angles are supplementary.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, between the parallel lines. The two angles are supplementary.
Consecutive exterior angles theorem

If a transversal intersects two parallel lines, then same-side (consecutive) exterior angles are supplementary.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, outside the parallel lines. The two angles are supplementary.
Alternate interior angles theorem

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, between the parallel lines. The two angles are congruent.
Alternate exterior angles theorem

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, outside the parallel lines. The two angles are congruent.

Examples

Example 1

For each of the following angle pairs, state the type of angle pair they are and state the relationship between the angles:

Two parallel lines intersected by a horizontal transversal. On one side of the transversal labeled from left to right, the angles are A, B, C, and an unlabeled angle. An angle labeled D is adjacent to angle B. An angle labeled E and angle C is a pair of vertical angles.
a

\angle A and \angle C

Worked Solution
Apply the idea

The angles \angle A and \angle C are corresponding angles formed by a transversal crossing a pair of parallel lines.

By the corresponding angles theorem, they are congruent.

Reflect and check

The angles \angle A and \angle C are not considered same-side (consecutive) interior or exterior angles because \angle A is exterior while \angle C is interior.

b

\angle B and \angle C

Worked Solution
Apply the idea

The angles \angle B and \angle C are same-side (consecutive) interior angles formed by a transversal crossing a pair of parallel lines.

By the consecutive interior angles theorem, they are supplementary.

Reflect and check

Recall that supplementary angles have a sum of 180 \degree.

c

\angle C and \angle D

Worked Solution
Create a strategy

Note that \angle C and \angle D are on opposite sides of the transversal, so we know they are alternating.

Apply the idea

The angles \angle C and \angle D are alternate interior angles formed by a transversal crossing a pair of parallel lines.

By the alternate interior angles theorem, they are congruent.

d

\angle A and \angle E

Worked Solution
Apply the idea

The angles \angle A and \angle E are alternate exterior angles formed by a transversal crossing a pair of parallel lines.

By the alternate exterior angles theorem, they are congruent.

Reflect and check

\angle A and \angle E would only be considered same-side (consecutive) if they were on the same side of the transversal.

Example 2

The figure shows two intersecting pairs of parallel lines.

A pair of parallel lines intersected by another pair of parallel horizontal lines. An angle labeled 63 degrees is formed by top and left lines. An angle labeled x degrees is formed by the left and bottom lines. The angle labeled 63 degrees, and the angle labeled x degrees is a pair of consecutive exterior angles. An angle labeled y degrees is formed by the right and bottom lines. The angle labeled x, and the angle labeled y degrees is a pair of alternate interior angles.
a

Find the value of x and explain your answer.

Worked Solution
Create a strategy

We can see that angle labeled with a measure of x \degree forms a same-side (consecutive) exterior angle pair with the given angle. Since they lie on the transversal of two parallel lines, we can use the consecutive exterior angle theorem to relate their measures.

Apply the idea

By the consecutive exterior angle theorem, the two angles are supplementary. This means that:

x + 63 = 180

Solving this equation tells us that x = 117.

b

Find the value of y and explain your answer.

Worked Solution
Create a strategy

We can see that the angles labeled with measures of x\degree and y\degree form a pair of alternate interior angles. Since they lie on the transversal of two parallel lines, we can use the alternate interior angles theorem to relate their measures.

Apply the idea

By the alternate interior angles theorem, the angles labeled with measures of x\degree and y\degree are congruent. This means that:

x=y

Using the value of x found in the previous part, this tells us that y = 117.

Reflect and check

When we have two intersecting pairs of parallel lines, we can use the theorems introduced in this topic to relate the measures of all the angles formed by their intersection.

Example 3

Determine if the information given is enough to justify the conclusion.

Given: a\parallel b and \angle 1 \cong \angle 3

Conclusion: \angle 2 and \angle 3 are supplementary

Lines a, b, and c intersected by a transversal. Lines a and b are parallel. An angle labeled 1 is formed by line a and the transversal. Angles labeled 4 and 2 are formed by line b and the transversal. An angle labeled 3 degrees is formed by line c and the transversal. Angles 1, 4, and 3 lie on the same side of the transversal, and on the same side of the lines a, b, and c respectively. Angle 2 lies on the opposite side of the transversal, and on the same side of line b as angle 4.
Worked Solution
Create a strategy

Use postulates and theorems about angles to determine if there is enough information to justify the conclusion.

Apply the idea

Yes, we can conclude \angle 2 and \angle 3 are supplementary.

Since a \parallel b, \angle 1 \cong \angle 4 by the corresponding angles theorem and \angle 4 and \angle 2 are supplementary by the linear pair postulate. So using the previous two statements, \angle 1 and \angle 2 are supplementary. Since \angle 1 \cong \angle 3, and \angle 1 and \angle 2 are supplementary, we can conclude that \angle 2 and \angle 3 are supplementary.

Example 4

Construct a proof of the following:

Given: \overleftrightarrow{AB} \parallel \overleftrightarrow{CD}

Prove: m \angle 5 = m \angle 3 and m \angle 1 = m \angle 7

Two horizontal lines A B and C D intersected by a transversal. At the intersection of A B and the transversal, labeled clockwise from top left, are angles 2, 1, 3, and 4. At the intersection of C D and the transversal, labeled clockwise from top left, are angles 6, 5, 8, and 7.
Worked Solution
Create a strategy

We can write a proof using the structure of a two column proof, paragraph proof, or flow chart proof.

Apply the idea

One approach to prove the conclusion is with a flow chart proof, as shown:

A flow chart proof with 5 levels and the reason below each step. The fourth and fifth levels have 2 steps. On the first level, the step is labeled line A B is parallel to line C D, with reason Given. On the second level, the step is labeled angle 1 is congruent to angle 5, with reason Corresponding angles theorem. On the third level, the step is labeled measure of angle 1 equals measure of angle 5, with reason Definition of congruent angles. On the fourth level, the left step is labeled measure of angle 1 equals measure of angle 3, with reason Vertical angles theorem, and the right step is labeled measure of angle 5 equals measure of angle 7, with reason Vertical angles theorem. On the fifth level, the left step is labeled measure of angle 5 equals measure of angle 3, with reason Transitive property of equality, and the right step is labeled measure of angle 1 equals measure of angle 7, with reason Transitive property of equality.
Reflect and check

Notice that we just proved the alternate interior angles theorem and the alternate exterior angles theorem.

Idea summary

When a transversal cuts through two parallel lines,

  • Corresponding angles are congruent

  • Same-side (consecutive) interior angles and Same-side (consecutive) exterior angles are supplementary

  • Alternate interior angles and alternate exterior angles are congruent

Outcomes

G.RLT.2

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

G.RLT.2a

Prove and justify angle pair relationships formed by two parallel lines and a transversal, including: i) corresponding angles; ii) alternate interior angles; iii) alternate exterior angles; iv) same-side (consecutive) interior angles; and v) same-side (consecutive) exterior angles.

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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