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2. Parallel & Perpendicular Lines
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United States of AmericaFL
Grade 912

2.02 Proving lines parallel

Lesson
Practice
Lesson

Concept summary

To determine if two lines are parallel, we can use the converses of the theorems which relate angle pairs formed by two lines and a transversal.

Converse of corresponding angles postulate

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

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.
Converse of consecutive interior angles theorem

If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.

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.
Converse of consecutive exterior angles theorem

If two lines and a transversal form consecutive exterior angles that are supplementary, then the lines are parallel.

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.
Converse of alternate interior angles theorem

If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

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.
Converse of alternate exterior angles theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.

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.

Since both the theorems and their converses are true, we can write each of the five statements as biconditional statements.

For example, for alternate exterior angles all of the following are true and can be used to solve problems.

  • Statement: If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
  • Converse: If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.
  • Biconditional statement: For two lines and a transversal, alternate exterior angles are congruent if and only if the lines are parallel.

Worked examples

Example 1

Determine whether or not there is a pair of parallel lines in the figure.

Approach

We can see that the two marked angles form a pair of consecutive interior angles. This means that we can use the converse of consecutive interior angles theorem to check whether or not we have parallel lines.

Solution

If we add the measures of the consecutive interior angles together we get:

124\degree+46\degree=170\degree

Since the sum of the measures is not 180\degree, the two angles are not supplementary.

The converse of consecutive interior angles states that the lines are parallel if the consecutive interior angles are supplementary. Since they are not supplementary, the lines are not parallel.

Therefore, there is not a pair of parallel lines in the figure.

Example 2

Find the value of x required for the figure to contain a pair of parallel lines.

Approach

We can see that the two marked angles form a pair of alternate exterior angles. For there to be a pair of parallel lines, the converse of alternate exterior angles theorem tells us that the two marked angles must be congruent.

Solution

We know that the two marked angles must be congruent, so we can set their measures to be equal and solve for x.

\displaystyle 5x-8\displaystyle =\displaystyle 3x+40Congruent angles have equal measures
\displaystyle 5x-3x\displaystyle =\displaystyle 40+8Add 8 and subtract 3x from both sides
\displaystyle 2x\displaystyle =\displaystyle 48Simplify
\displaystyle x\displaystyle =\displaystyle 24Divide both sides by 2

Therefore, the figure contains a pair of parallel lines when x=24.

Reflection

For any other value of x, the equality would not be true and the lines would not be parallel.

Example 3

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

Given: \angle 1 \cong \angle 3

Conclusion: a \parallel c

Solution

No, we do not have enough information to conclude a \parallel c.

Given that \angle 1 \cong \angle 3, we could conclude that a \parallel b, using the converse of corresponding angles postulate. However, we do not have enough information to the conclude that a \parallel c. We would need some information involving \angle 4 to draw a conclusion.

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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