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# 3.2: Properties of Parallel Lines

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Determine what happens to corresponding angles, alternate interior angles, alternate exterior angles, and same side interior angles when two lines are parallel.

## Review Queue

Use the picture below to determine:

1. A pair of corresponding angles.
2. A pair of alternate interior angles.
3. A pair of alternate exterior angles.
4. A pair of same side interior angles.

Know What? The streets below are in Washington DC. The red street and the blue street are parallel. The transversals are the green and orange streets.

1. If \begin{align*}m\angle FTS = 35^\circ\end{align*}, determine the other angles that are \begin{align*}35^\circ\end{align*}.
2. If \begin{align*}m\angle SQV = 160^\circ\end{align*}, determine the other angles that are \begin{align*}160^\circ\end{align*}.

## Corresponding Angles Postulate

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

If \begin{align*}l || m\end{align*}, then \begin{align*}\angle 1 \cong \angle 2\end{align*}.

Example 1: If \begin{align*}a || b\end{align*}, which pairs of angles are congruent by the Corresponding Angles Postulate?

Solution: There are 4 pairs of congruent corresponding angles:

\begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}, and \begin{align*}\angle 4 \cong \angle 8\end{align*}.

Investigation 3-4: Corresponding Angles Exploration

1. Place your ruler on the paper. On either side of the ruler, draw 2 lines, 3 inches long.

This is the easiest way to ensure that the lines are parallel.

2. Remove the ruler and draw a transversal. Label the eight angles as shown.

3. Using your protractor, measure all of the angles. What do you notice?

You should notice that all the corresponding angles have equal measures.

Example 2: If \begin{align*}m\angle 2 = 76^\circ\end{align*}, what is \begin{align*}m\angle 6\end{align*}?

Solution: \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} are corresponding angles and \begin{align*}l || m\end{align*} from the arrows on them. \begin{align*}\angle 2 \cong \angle 6\end{align*} by the Corresponding Angles Postulate, which means that \begin{align*}m\angle 6 = 76^\circ\end{align*}.

Example 3: Using the measures of \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} from Example 2, find all the other angle measures.

Solution: If \begin{align*}m\angle 2 = 76^\circ\end{align*}, then \begin{align*}m\angle 1 = 180^\circ - 76^\circ =104^\circ\end{align*} (linear pair). \begin{align*}\angle 3 \cong \angle 2\end{align*} (vertical angles), so \begin{align*}m\angle 3 = 76^\circ. \ m\angle 4 = 104^\circ\end{align*} (vertical angle with \begin{align*}\angle 1\end{align*}).

By the Corresponding Angles Postulate, we know \begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}, and \begin{align*}\angle 4 \cong \angle 8\end{align*}, so \begin{align*}m\angle 5 = 104^\circ, \ m\angle 6 = 76^\circ, \ m\angle 7 = 76^\circ\end{align*}, and \begin{align*}m\angle 104^\circ\end{align*}.

## Alternate Interior Angles Theorem

Example 4: Find \begin{align*}m\angle 1\end{align*}.

Solution: \begin{align*}m\angle 2 = 115^\circ\end{align*} because they are corresponding angles and the lines are parallel. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are vertical angles, so \begin{align*}m\angle 1 = 115^\circ\end{align*}.

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

If \begin{align*}l || m\end{align*}, then \begin{align*}\angle 1 \cong \angle 2\end{align*}

Proof of Alternate Interior Angles Theorem

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

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

Statement Reason
1. \begin{align*}l || m\end{align*} Given
2. \begin{align*}\angle 3 \cong \angle 7\end{align*} Corresponding Angles Postulate
3. \begin{align*}\angle 7 \cong \angle 6\end{align*} Vertical Angles Theorem
4. \begin{align*}\angle 3 \cong \angle 6\end{align*} Transitive \begin{align*}PoC\end{align*}

We could have also proved that \begin{align*}\angle 4 \cong \angle 5\end{align*}.

Example 5: Algebra Connection Find the measure of \begin{align*}x\end{align*}.

Solution: The two given angles are alternate interior angles and equal.

\begin{align*}(4x-10)^\circ & = 58^\circ\\ 4x & = 68^\circ\\ x & = 17^\circ\end{align*}

Alternate Exterior Angles Theorem

Example 6: Find \begin{align*}m\angle 1\end{align*} and \begin{align*}m\angle 2\end{align*}.

Solution: \begin{align*}m\angle 1 = 47^\circ\end{align*} by vertical angles. The lines are parallel, so \begin{align*}m\angle 2 = 47^\circ\end{align*} by the Corresponding Angles Postulate.

Here, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are alternate exterior angles.

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

If \begin{align*}l || m\end{align*}, then \begin{align*}\angle 1 \cong \angle 2\end{align*}.

Example 7: Algebra Connection Find the measure of each angle and the value of \begin{align*}y\end{align*}.

Solution: The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

\begin{align*}(3y+53)^\circ & = (7y-55)^\circ\\ 108^\circ & = 4y\\ 27^\circ & = y\end{align*}

If \begin{align*}y = 27^\circ\end{align*}, then each angle is \begin{align*}3(27^\circ) + 53^\circ = 134^\circ\end{align*}.

Same Side Interior Angles Theorem

Same side interior angles are on the interior of the parallel lines and on the same side of the transversal. They have a different relationship that the other angle pairs.

Example 8: Find \begin{align*}m\angle 2\end{align*}.

Solution: \begin{align*}\angle 1\end{align*} and \begin{align*}66^\circ\end{align*} are alternate interior angles, so \begin{align*}m\angle 1 = 66^\circ\end{align*}. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair, so they add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}m\angle 1+m\angle 2 & = 180^\circ\\ 66^\circ + m\angle 2 & = 180^\circ\\ m\angle 2 & = 114^\circ\end{align*}

This example shows that if two parallel lines are cut by a transversal, the same side interior angles add up to \begin{align*}180^\circ\end{align*}.

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

If \begin{align*}l || m\end{align*}, then \begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}.

Example 9: Find \begin{align*}x, y\end{align*}, and \begin{align*}z\end{align*}.

Solution: \begin{align*}x = 73^\circ\end{align*} by Alternate Interior Angles

\begin{align*}y = 107^\circ\end{align*} because it is a linear pair with \begin{align*}x\end{align*}.

\begin{align*}z = 64^\circ\end{align*} by Same Side Interior Angles.

Example 10: Algebra Connection Find the measure of \begin{align*}x\end{align*}.

Solution: The given angles are same side interior angles. Because the lines are parallel, the angles add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}(2x+43)^\circ + (2x-3)^\circ & = 180^\circ\\ (4x+40)^\circ & = 180^\circ\\ 4x & = 140^\circ\\ x & =35^\circ\end{align*}

Example 11: \begin{align*}l || m\end{align*} and \begin{align*}s || t\end{align*}. Explain how \begin{align*}\angle 1 \cong \angle 16\end{align*}.

Solution: Because \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 16\end{align*} are not on the same transversal, we cannot assume they are congruent.

\begin{align*}\angle 1 \cong \angle 3\end{align*} by Corresponding Angles

\begin{align*}\angle 3 \cong \angle 16\end{align*} by Alternate Exterior Angles

\begin{align*}\angle 1 \cong \angle 16\end{align*} by the Transitive Property

Know What? Revisited Using what we have learned in this lesson, the other angles that are \begin{align*}35^\circ\end{align*} are \begin{align*}\angle TLQ, \ \angle ETL\end{align*}, and the vertical angle with \begin{align*}\angle TLQ\end{align*}. The other angles that are \begin{align*}160^\circ\end{align*} are \begin{align*}\angle FSR, \ \angle TSQ\end{align*}, and the vertical angle with \begin{align*}\angle SQV\end{align*}.

## Review Questions

• Questions 1-7 use the theorems learned in this section.
• Questions 8-16 are similar to Example 11.
• Questions 17-20 are similar to Example 6, 8 and 9.
• Questions 21-25 are similar to Examples 5, 7, and 10.
• Questions 26-29 are similar to the proof of the Alternate Interior Angles Theorem.
• Question 30 uses the theorems learned in this section.

For questions 1-7, determine if each angle pair below is congruent, supplementary or neither.

1. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 7\end{align*}
2. \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 2\end{align*}
3. \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 3\end{align*}
4. \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 8\end{align*}
5. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 6\end{align*}
6. \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 6\end{align*}
7. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*}

For questions 8-16, determine if the angle pairs below are: Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Same Side Interior Angles, Vertical Angles, Linear Pair or None.

1. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 13\end{align*}
2. \begin{align*}\angle 7\end{align*} and \begin{align*}\angle 12\end{align*}
3. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 11\end{align*}
4. \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 10\end{align*}
5. \begin{align*}\angle 14\end{align*} and \begin{align*}\angle 9\end{align*}
6. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 11\end{align*}
7. \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 15\end{align*}
8. \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 16\end{align*}
9. List all angles congruent to \begin{align*}\angle 8\end{align*}.

For 17-20, find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

Algebra Connection For questions 21-25, use the picture to the right. Find the value of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

1. \begin{align*}m\angle 1 = (4x + 35)^\circ, \ m\angle 8 = (7x - 40)^\circ\end{align*}
2. \begin{align*}m\angle 2 = (3y + 14)^\circ, \ m\angle 6 = (8x - 76)^\circ\end{align*}
3. \begin{align*}m\angle 3 = (3x +12)^\circ, \ m\angle 5 = (5x + 8)^\circ\end{align*}
4. \begin{align*}m\angle 4 = (5x - 33)^\circ, \ m\angle 5 = (2x + 60)^\circ\end{align*}
5. \begin{align*}m\angle 1 = (11y - 15)^\circ, \ m \angle 7 = (5y + 3)^\circ\end{align*}

Fill in the blanks in the proofs below.

1. Given: \begin{align*}l || m\end{align*} Prove: \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are supplementary (Same Side Interior Angles Theorem)
Statement Reason
1. Given
2. \begin{align*}\angle 1 \cong \angle 5\end{align*}
3. \begin{align*}\cong\end{align*} angles have = measures
4. Linear Pair Postulate
5. Definition of Supplementary Angles
6. \begin{align*}m\angle 3 + m\angle 5 = 180^\circ\end{align*}
7. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are supplementary
1. Given: \begin{align*}l || m\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 8\end{align*} (Alternate Exterior Angles Theorem)
Statement Reason
1.
2. \begin{align*}\angle 1 \cong \angle 5\end{align*}
3. Vertical Angles Theorem
4. \begin{align*}\angle 1 \cong \angle 8\end{align*}

For 28 and 29, use the picture to the right.

1. Given: \begin{align*}l || m, \ s || t\end{align*} Prove: \begin{align*}\angle 2 \cong \angle 15\end{align*}
Statement Reason
1. \begin{align*}l || m, \ s || t\end{align*}
2. \begin{align*}\angle 2 \cong \angle 13\end{align*}
3. Corresponding Angles Postulate
4. \begin{align*}\angle 2 \cong \angle 15\end{align*}
1. Given: \begin{align*}l || m, \ s || t\end{align*} Prove: \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 9\end{align*} are supplementary
Statement Reason
1.
2. \begin{align*}\angle 6 \cong \angle 9\end{align*}
3. \begin{align*}\angle 4 \cong \angle 7\end{align*}
4. Same Side Interior Angles
5. \begin{align*}\angle 9\end{align*} an \begin{align*}\angle 4\end{align*} are supplementary
1. Find the measures of all the numbered angles in the figure below.

## Review Queue Answers

1. \begin{align*}\angle {1}\end{align*} and \begin{align*}\angle {6}, \angle {2}\end{align*} and \begin{align*}\angle {8}, \angle {3}\end{align*} and \begin{align*}\angle {7}\end{align*}, or \begin{align*}\angle {4}\end{align*} and \begin{align*}\angle {5}\end{align*}
2. \begin{align*}\angle {2}\end{align*} and \begin{align*}\angle {5}\end{align*} or \begin{align*}\angle {3}\end{align*} and \begin{align*}\angle {6}\end{align*}
3. \begin{align*}\angle {1}\end{align*} and \begin{align*}\angle {7}\end{align*} or \begin{align*}\angle {4}\end{align*} and \begin{align*}\angle {8}\end{align*}
4. \begin{align*}\angle {3}\end{align*} and \begin{align*}\angle {5}\end{align*} or \begin{align*}\angle {2}\end{align*} and \begin{align*}\angle {6}\end{align*}

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CK.MAT.ENG.SE.1.Geometry-Basic.3.2