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# 3.3: Proving Lines Parallel

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use the converses of the Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, and the Consecutive Interior Angles Theorem to show that lines are parallel.
• Construct parallel lines using the above converses.
• Use the Parallel Lines Property.

## Review Queue

1. Write the converse of the following statements:
1. If it is summer, then I am out of school.
2. I will go to the mall when I am done with my homework.
2. Are any of the converses from #1 true? Give a counterexample, if not.
3. Determine the value of \begin{align*}x\end{align*} if \begin{align*}l || m\end{align*}.
4. What is the measure of each angle in #3?

Know What? Here is a picture of the support beams for the Coronado Bridge in San Diego. To aid the strength of the curved bridge deck, the support beams should not be parallel.

This bridge was designed so that \begin{align*}\angle 1 = 92^\circ\end{align*} and \begin{align*}\angle 2 = 88^\circ\end{align*}. Are the support beams parallel?

## Corresponding Angles Converse

Recall that the converse of

If \begin{align*}a\end{align*}, then \begin{align*}b\end{align*} is

If \begin{align*}b\end{align*}, then \begin{align*}a\end{align*}

For the Corresponding Angles Postulate:

If two lines are parallel, then the corresponding angles are congruent.

BECOMES

If corresponding angles are congruent, then the two lines are parallel.

Is this true? If corresponding angles are both \begin{align*}60^\circ\end{align*}, would the lines be parallel?

If then, is \begin{align*}l || m\end{align*}?

YES. Congruent corresponding angles make the slopes of \begin{align*}l\end{align*} and \begin{align*}m\end{align*} the same which makes the lines parallel.

Investigation 3-5: Creating Parallel Lines using Corresponding Angles

1. Draw two intersecting lines. Make sure they are not perpendicular. Label them \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, and the point of intersection, \begin{align*}A\end{align*}, as shown.

2. Create a point, \begin{align*}B\end{align*}, on line \begin{align*}m\end{align*}, above \begin{align*}A\end{align*}.

3. Copy the acute angle at \begin{align*}A\end{align*} (the angle to the right of \begin{align*}m\end{align*}) at point \begin{align*}B\end{align*}. See Investigation 2-2 in Chapter 2 for the directions on how to copy an angle.

4. Draw the line from the arc intersections to point \begin{align*}B\end{align*}.

The copied angle allows the line through point \begin{align*}B\end{align*} to have the same slope as line \begin{align*}l\end{align*}, making the two lines parallel.

Converse of Corresponding Angles Postulate: If corresponding angles are congruent when two lines are cut by a transversal, then the lines are parallel.

If then \begin{align*}l || m\end{align*}.

Example 1: If \begin{align*}m\angle 8 = 110^\circ\end{align*} and \begin{align*}m\angle 4 = 110^\circ\end{align*}, then what do we know about lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*}?

Solution: \begin{align*}\angle 8\end{align*} and \begin{align*}\angle 4\end{align*} are corresponding angles. Since \begin{align*}m\angle 8 = m\angle 4\end{align*}, we can conclude that \begin{align*}l || m\end{align*}.

Example 2: Is \begin{align*}l || m\end{align*}?

Solution: The two angles are corresponding and must be equal to say that \begin{align*}l || m\end{align*}. \begin{align*}116^\circ \neq 118^\circ\end{align*}, so \begin{align*}l\end{align*} is not parallel to \begin{align*}m\end{align*}.

## Alternate Interior Angles Converse

The converse of the Alternate Interior Angles Theorem is:

Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

If then \begin{align*}l || m\end{align*}.

Example 3: Prove the Converse of the Alternate Interior Angles Theorem.

Given: \begin{align*}l\end{align*} and \begin{align*}m\end{align*} and transversal \begin{align*}t\end{align*}

\begin{align*}\angle 3 \cong \angle 6\end{align*}

Prove: \begin{align*}l||m\end{align*}

Solution:

Statement Reason
1. \begin{align*}l\end{align*} and \begin{align*}m\end{align*} and transversal \begin{align*}t\end{align*} \begin{align*}\angle 3 \cong \angle 6\end{align*} Given
2. \begin{align*}\angle 3 \cong \angle 2\end{align*} Vertical Angles Theorem
3. \begin{align*}\angle 2 \cong \angle 6\end{align*} Transitive \begin{align*}PoC\end{align*}
4. \begin{align*}l||m\end{align*} Converse of the Corresponding Angles Postulate

Prove Move: Shorten the names of these theorems. For example, the Converse of the Corresponding Angles Postulate could be “Conv CA Post.”

Example 4: Is \begin{align*}l || m\end{align*}?

Solution: Find \begin{align*}m\angle 1\end{align*}. We know its linear pair is \begin{align*}109^\circ\end{align*}, so they add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}m\angle 1 + 109^\circ & = 180^\circ\\ m\angle 1 & = 71^\circ.\end{align*}

This means \begin{align*}l || m\end{align*}.

Example 5: Algebra Connection What does \begin{align*}x\end{align*} have to be to make \begin{align*}a || b\end{align*}?

Solution: The angles are alternate interior angles, and must be equal for \begin{align*}a || b\end{align*}. Set the expressions equal to each other and solve.

\begin{align*}3x+16^\circ & = 5x-54^\circ\\ 70^\circ & = 2x\\ 35^\circ & = x\end{align*}

To make \begin{align*}a || b, \ x = 35^\circ\end{align*}.

Converse of Alternate Exterior Angles & Consecutive Interior Angles

You have probably guessed that the converse of the Alternate Exterior Angles Theorem and the Consecutive Interior Angles Theorem are true.

Converse of the Alternate Exterior Angles Theorem: If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.

If then \begin{align*}l || m\end{align*}.

Example 6: Real-World Situation The map below shows three roads in Julio’s town.

Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.

Solution: The \begin{align*}130^\circ\end{align*} angle and \begin{align*}\angle a\end{align*} are alternate exterior angles. If \begin{align*}m\angle a = 130^\circ\end{align*}, then the lines are parallel.

\begin{align*}\angle a + 40^\circ & = 180^\circ && \text{by the Linear Pair Postulate}\\ \angle a & = 140^\circ\end{align*}

\begin{align*}140^\circ \neq 130^\circ\end{align*}, so Franklin Way and Chavez Avenue are not parallel streets.

The final converse theorem is the Same Side Interior Angles Theorem. Remember that these angles aren’t congruent when lines are parallel, they add up to \begin{align*}180^\circ\end{align*}.

Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.

If then \begin{align*}l || m\end{align*}.

Example 7: Is \begin{align*}l || m\end{align*}? How do you know?

Solution: These angles are Same Side Interior Angles. So, if they add up to \begin{align*}180^\circ\end{align*}, then \begin{align*}l || m\end{align*}.

\begin{align*}113^\circ + 67^\circ = 180^\circ\end{align*}, therefore \begin{align*}l || m\end{align*}.

## Parallel Lines Property

The Parallel Lines Property is a transitive property for parallel lines. The Transitive Property of Equality is: If \begin{align*}a = b\end{align*} and \begin{align*}b = c\end{align*}, then \begin{align*}a = c\end{align*}. The Parallel Lines Property changes = to \begin{align*}||\end{align*}.

Parallel Lines Property: If lines \begin{align*}l || m\end{align*} and \begin{align*}m || n\end{align*}, then \begin{align*}l || n\end{align*}.

If then

Example 8: Are lines \begin{align*}q\end{align*} and \begin{align*}r\end{align*} parallel?

Solution: First find if \begin{align*}p || q\end{align*}, then \begin{align*}p || r\end{align*}. If so, \begin{align*}q || r\end{align*}.

\begin{align*}p || q\end{align*} because the corresponding angles are equal.

\begin{align*}p || r\end{align*} because the alternate exterior angles are equal.

\begin{align*}q || r\end{align*} by the Parallel Lines Property.

Know What? Revisited: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are corresponding angles and must be equal for the beams to be parallel. \begin{align*}\angle 1 = 92^\circ\end{align*} and \begin{align*}\angle 2 = 88^\circ\end{align*}, so they are not equal and the beams are not parallel, therefore the bridge is study and safe.

## Review Questions

• Questions 1-13 are similar to Examples 1, 2, 4, and 7.
• Question 14 is similar to Investigation 3-1.
• Question 15 uses the Corresponding Angles Postulate and its converse.
• Questions 16-22 are similar to Example 3.
• Questions 23-36 are similar to Examples 1, 2, 4, and 7.
• Questions 37-40 are similar to Example 5.
1. Are lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} parallel? If yes, how do you know?
2. Are lines 1 and 2 parallel? Why or why not?
3. Are the lines below parallel? Why or why not?

Use the following diagram. \begin{align*}m || n\end{align*} and \begin{align*}p \perp q\end{align*}. Find each angle and give a reason for each answer.

1. \begin{align*}a = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}b = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}d = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}e = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}f = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}g = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}h = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. Construction Using Investigation 3-1 to help you, show that two lines are parallel by constructing congruent alternate interior angles. HINT: Steps 1 and 2 will be the same, but at step 3, you will copy the angle in a different spot.
10. Writing Explain when you would use the Corresponding Angles Postulate and the Converse of the Corresponding Angles Postulate in a proof.

For Questions 16-22, fill in the blanks in the proofs below.

1. Given: \begin{align*}l || m, \ p || q\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 2\end{align*}
Statement Reason
1. \begin{align*}l || m\end{align*} 1.
2. 2. Corresponding Angles Postulate
3. \begin{align*}p || q\end{align*} 3.
4. 4.
5. \begin{align*}\angle 1 \cong \angle 2\end{align*} 5.
1. Given: \begin{align*}p || q, \ \angle 1 \cong \angle 2\end{align*} Prove: \begin{align*}l || m\end{align*}
Statement Reason
1. \begin{align*}p || q\end{align*} 1.
2. 2. Corresponding Angles Postulate
3. \begin{align*}\angle 1 \cong \angle 2\end{align*} 3.
4. 4. Transitive \begin{align*}PoC\end{align*}
5. 5. Converse of Alternate Interior Angles Theorem
1. Given: \begin{align*}\angle 1 \cong \angle 2, \ \angle 3 \cong \angle 4\end{align*} Prove: \begin{align*}l || m\end{align*}
Statement Reason
1. \begin{align*}\angle 1 \cong \angle 2\end{align*} 1.
2. \begin{align*}l || n\end{align*} 2.
3. \begin{align*}\angle 3 \cong \angle 4\end{align*} 3.
4. 4. Converse of Alternate Interior Angles Theorem
5. \begin{align*}l || m\end{align*} 5.
1. Given: \begin{align*}m \perp l, \ n \perp l\end{align*} Prove: \begin{align*}m||n\end{align*}
Statement Reason
1. \begin{align*}m \perp l, \ n \perp l\end{align*}
2. \begin{align*}m\angle 1= 90^\circ, \ m\angle 2=90^\circ\end{align*}
3. Transitive Property
4. \begin{align*}m || n\end{align*}
1. Given: \begin{align*}\angle 1 \cong \angle 3\end{align*} Prove: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*} are supplementary
Statement Reason
1.
2. \begin{align*}m || n\end{align*}
3. Linear Pair Postulate
4. Substitution
5. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*} are supplementary
1. Given: \begin{align*}\angle 2 \cong \angle 4\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 3\end{align*}
Statement Reason
1.
2. \begin{align*}m || n\end{align*}
3. \begin{align*}\angle 1 \cong \angle 3\end{align*}
1. Given: \begin{align*}\angle 2 \cong \angle 3\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 4\end{align*}
Statement Reason
1.
2. Converse of Corresponding Angles Theorem
3. \begin{align*}\angle 1 \cong \angle 4\end{align*}

In 23-29, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

1. \begin{align*}\angle BDC \cong \angle JIL\end{align*}
2. \begin{align*}\angle AFD\end{align*} and \begin{align*}\angle BDF\end{align*} are supplementary
3. \begin{align*}\angle EAF \cong \angle FJI\end{align*}
4. \begin{align*}\angle EFJ \cong \angle FJK\end{align*}
5. \begin{align*}\angle DIE \cong \angle EAF\end{align*}
6. \begin{align*}\angle EDB \cong \angle KJM\end{align*}
7. \begin{align*}\angle DIJ\end{align*} and \begin{align*}\angle FJI\end{align*} are supplementary

In 30-36, find the measure of the lettered angles below.

1. \begin{align*}m\angle 1\end{align*}
2. \begin{align*}m\angle 2\end{align*}
3. \begin{align*}m\angle 3\end{align*}
4. \begin{align*}m\angle 4\end{align*}
5. \begin{align*}m\angle 5\end{align*}
6. \begin{align*}m\angle 6\end{align*}
7. \begin{align*}m\angle 7\end{align*}

Algebra Connection For 37-40, what does \begin{align*}x\end{align*} have to measure to make the lines parallel?

1. \begin{align*}m\angle 3 = (3x+25)^\circ\end{align*} and \begin{align*}m\angle 5 = (4x-55)^\circ\end{align*}
2. \begin{align*}m\angle 2 = (8x)^\circ\end{align*} and \begin{align*}m\angle 7 = (11x-36)^\circ\end{align*}
3. \begin{align*}m\angle 1 = (6x-5)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x+7)^\circ\end{align*}
4. \begin{align*}m\angle 4 = (3x-7)^\circ\end{align*} and \begin{align*}m\angle 7 = (5x-21)^\circ\end{align*}

1. If I am out of school, then it is summer.
2. If I go to the mall, then I am done with my homework.
1. Not true, I could be out of school on any school holiday or weekend during the school year.
2. Not true, I don’t have to be done with my homework to go to the mall.
1. The two angles are supplementary. \begin{align*}(17x + 14)^\circ + (4x - 2)^\circ &= 180^\circ\\ 21x + 12^\circ &= 180^\circ\\ 21x &= 168^\circ\\ x &= 8^\circ\end{align*}
2. The angles are \begin{align*}17(8^\circ) + 14^\circ = 150^\circ\end{align*} and \begin{align*}180^\circ - 150^\circ = 30^\circ\end{align*}

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