1.4.1: Alternate Interior Angles

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Illustrative Mathematics
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Lesson

Let's explore why some angles are always equal.

Exercise \(\PageIndex{1}\): Angle Pairs

1. Find the measure of angle \(JGH\). Explain or show your reasoning.

2. Find and label a second \(30^{\circ}\) degree angle in the diagram. Find and label an angle congruent to angle \(JGH\).

Exercise \(\PageIndex{2}\): Cutting Parallel Lines with a Transversal

Lines \(AC\) and \(DF\) are parallel. They are cut by transversal \(HJ\).

Figure \(\PageIndex{2}\): Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled question mark. Angle F E B is labeled question mark. Angle B E D is labeled question mark.

With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.
What do you notice about the angles with vertex \(B\) and the angles with vertex \(E\)?
Using what you noticed, find the measures of the four angles at point \(B\) in the second diagram. Lines \(AC\) and \(DF\) are parallel.

Figure \(\PageIndex{3}\): Line A C contains point B. Line D F contains point E. Line H G contains points B and E. Angle A B H is labeled with a question mark. Angle A B E is labeled with a question mark. Angle E B C is labeled with a question mark. Angle C B H is labeled with a question mark. Angle G E F is labeled 34 degrees.

4. The next diagram resembles the first one, but the lines form slightly different angles. Work with your partner to find the six unknown angles with vertices at points \(B\) and \(E\).

Figure \(\PageIndex{4}\): Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled 108 degrees. Angle F E B is labeled question mark. Angle B E D is labeled question mark.

5. What do you notice about the angles in this diagram as compared to the earlier diagram? How are the two diagrams different? How are they the same?

Are you ready for more?

Figure \(\PageIndex{5}\): Four lines on a grid. Line l is a horizontal line 8 units up on the grid. Line m is a horizontal line 3 units up on the grid. A line trends up and to the right and crossed line m at a point between 3 and 4 units to the right and 3 units up. The line crosses line l at a point 7 units to the right and 8 units up. Another line crosses line m at a point between 9 and 10 units to the right and 3 units up. It crosses the line l at a point 7 units to the right and 8 units up. The interior angle at the top of the triangle formed is labeled 60 degrees. The bottom left exterior angle of the bottom left vertex is labeled 55 degrees. The bottom right exterior angle of the bottom right vertex is labeled x degrees.

Parallel lines \(l\) and \(m\) are cut by two transversals which intersect \(l\) in the same point. Two angles are marked in the figure. Find the measure \(x\) of the third angle.

Exercise \(\PageIndex{3}\): Alternate Interior Angles are Congruent

1. Lines \(l\) and \(k\) are parallel and \(t\) is a transversal. Point \(M\) is the midpoint of segment \(PQ\).

Find a rigid transformation showing that angles \(MPA\) and \(MQB\) are congruent.

2. In this picture, lines \(l\) and \(k\) are no longer parallel. \(M\) is still the midpoint of segment \(PQ\).

Does your argument in the earlier problem apply in this situation? Explain.

Summary

When two lines intersect, vertical angles are equal and adjacent angles are supplementary, that is, their measures sum to 180\(^{\circ}\). For example, in this figure angles 1 and 3 are equal, angles 2 and 4 are equal, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

Figure \(\PageIndex{8}\): Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.

When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles are created. (“Interior” means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.

Figure \(\PageIndex{9}\): Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.

Alternate interior angles are equal because a \(180^{\circ}\) rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point \(M\) halfway between the two intersections—can you see how rotating \(180^{\circ}\) about \(M\) takes angle 3 to angle 5?

Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is \(70^{\circ}\) we use vertical angles to see that angle 3 is \(70^{\circ}\), then we use alternate interior angles to see that angle 5 is \(70^{\circ}\), then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is \(110^{\circ}\) since \(180-70=110\). It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure \(70^{\circ}\), and angles 2, 4, 6, and 8 measure \(110^{\circ}\).

Glossary Entries

Definition: Alternate Interior Angles

Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

This diagram shows two pairs of alternate interior angles. Angles \(a\) and \(d\) are one pair and angles \(b\) and \(c\) are another pair.

Figure \(\PageIndex{10}\): There are two horizontal parallel lines, and a third diagonal line drawn from the bottom left to the upper right, intersecting both horizontal lines. The diagonal line is labeled transversal. There are four angles created by the diagonal line inside the parallel lines. The upper left angle is labeled a, upper right is b, lower left is c, and lower right is d.

Definition: Transversal

A transversal is a line that crosses parallel lines.

This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(l\).

Practice

Exercise \(\PageIndex{4}\)

Use the daigram to find the measure of each angle.

\(m\angle ABC\)
\(m\angle EBD\)
\(m\angle ABE\)

Figure \(\PageIndex{12}\): A horizontal line and a line that slopes downward from left to right. The lines intersect at a point labeled B. On the horizontal line, point E is to the left of B and C is to the right of B. On the sloping line, point A is above B and point D is below B. Angle C B D is labeled 45 degrees.

(From Unit 1.2.3)

Exercise \(\PageIndex{5}\)

Lines \(k\) and \(l\) are parallel, and the measure of angle \(ABC\) is 19 degrees.

Figure \(\PageIndex{13}\): Two parallel lines, k and l, cut by transversal line m. Points F, C and D are on line k. Points A and B are on line l. Points E, C and B are on transversal m. Angle E C F is marked congruent to angle D C B and to angle A B C.

Explain why the measure of angle \(ECF\) is 19 degrees. If you get stuck, consider translating line by moving \(B\) to \(C\).
What is the measure of angle \(BCD\)? Explain.

Exercise \(\PageIndex{6}\)

The diagram shows three lines with some marked angle measures.

Figure \(\PageIndex{14}\): Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angles are marked starting at the top right and going in the clockwise direction, 70 degrees, question mark, question mark, question mark. At the second intersection, angles are marked starting at the top right and going in the clockwise direction, 53 degrees, question mark, question mark, question mark.

Find the missing angle measures marked with question marks.

Exercise \(\PageIndex{7}\)

Lines \(s\) and \(t\) are parallel. Find the value of \(x\).

Exercise \(\PageIndex{8}\)

The two figures are scaled copies of each other.

What is the scale factor that takes Figure 1 to Figure 2?
What is the scale factor that takes Figure 2 to Figure 1?