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9. Properties of Lines & Triangles
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Integrated Math 1

9.01 Properties of parallel lines

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Concept summary

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.
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.
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.

We pair up the angles in this way because the following postulate and theorems tell us that they are either form supplementary or congruent pairs when the transversal cuts through two parallel lines.

Corresponding angles postulate

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.
Consecutive interior angles theorem

If a transversal intersects two parallel lines, then 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 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.

Worked examples

Example 1

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

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

Solution

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

By the corresponding angles postulate, they are congruent.

b

\angle B and \angle C

Solution

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

By the consecutive interior angles theorem, they are supplementary.

c

\angle C and \angle D

Solution

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

Solution

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.

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.

Approach

We can see that angle labeled with a measure of x\degree forms a 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.

Solution

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.

Approach

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.

Solution

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.

Reflection

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 given information is enough to justify the conclusion.

Given information: 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.

Solution

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

Since a \parallel b, \angle 1 \cong \angle 4 by the corresponding angles postulate 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.

Outcomes

M1.G.CO.B.3

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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