What transformation proves alternate interior angles are congruent?

Sometimes geometry feels like a giant parts warehouse. You trade a lot of number-crunching (not much addition, multiplication, subtraction or division in geometry) for a lot of inventory.

Parts of an Angle

For example, let's construct angle Z. We almost never write "angle Z," using instead a quick shorthand, ∠Z. Something as simple as an angle has parts.

1. Two rays, ZA and ZU, meet at Point Z
2. Where they meet at Point Z, they form a vertex, ∠Z

We say rays ZA and ZU, but those rays could also be small snippets out of longer lines that intersected at Point Z. They could be snippets cut as rays or as line segments, depending on taking an infinite chunk or a finite chunk of the infinite, intersecting lines.

Parallel Lines

Unlike the intersecting rays ZA and ZU, parallel lines never meet. The two lines, line segments, or rays never converge (move closer) or diverge (move away). The only sneaky way to get an angle from parallel lines is to declare each line is a straight angle, with a measure of 180°. While two points determine a line, if you locate three points on a line, you have created a straight angle with the middle point as the vertex.

Transversals

Parallel lines can be intersected by transversals. Any line cutting across parallel lines is a transversal. It can cross at any angle. A transversal intersecting parallel lines at 90° is perpendicular.

Alternate Interior Angles Definition

When a transversal intersects parallel lines, it creates an interior and exterior. Think about it: if you were two-dimensional and came across a line in your path, that line would stretch infinitely in two directions and you could not get past it. You would be outside, at the exterior, of the parallel lines. Just beyond the line and between it and the parallel line next to it, is the interior.

When the transversal intersects, it creates four angles at each parallel line, or eight angles altogether. Four of those angles are exterior and four are interior. We are interested in the four interior lines, those are our Alternate Interior Angles.

Let's create parallel lines LI and ON, and a transversal HE.

The two points where HE crosses the parallel lines are Points A and R. Yes, we have a HARE crossing a LION.

You mark drawings of parallel lines with little bird-feet marks, like Vs on their sides. Notice we have four exterior angles:

We have four interior angles:

We are only interested in the four interior angles. Two of the interior angles are built using the top parallel line LI, and two are built using the bottom parallel line, ON.

Alternate interior angles are two congruent angles from different parallel lines (one from LI, one from ON). This says:

• ∠LAR is an alternate interior angle with ∠ARN
• ∠IAR is an alternate interior angle with ∠ARO.

Alternate Interior Angles Theorem

Once you can recognize and break apart the various parts of parallel lines with transversals, you can use the Alternate Interior Angles Theorem to speed up your work.

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Converse of Alternate Interior Angles Theorem

The converse of the theorem is also true:

The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Alternate Interior Angles Examples

We can prove both these theorems so you can add them to your toolbox. With our origional figure, LI and ON are parallel lines (given) transversed by HE (given).

We could declare all sorts of relationships, but the proof can be short and simple:

• LI ∥ ON (given)
• ∠LAR ≅ ∠ORE (Corresponding Angles Postulate)
• ∠ORE ≅ ∠ARN (Vertical Angles Theorem)
• ∠LAR ≅ ∠ARN (Transitive Property of Congruence)

The Transitive Property of Congruence says if A is like B and B is like C, then A is like C. Since ∠LAR was congruent to ∠ORE, and ∠ORE was congruent to ∠ARN, then:

• ∠LAR ≅ ∠ORE ≅ ∠ARN
• ∠LAR ≅ ∠ARN

To prove the converse, "If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel," we work the other way around:

• LI and ON with transversal HE (given)
• ∠LAR ≅ ∠ARN (given)
• ∠LAR ≅ ∠HAI (Vertical Angles Theorem)
• ∠HAI ≅ ∠ARN (Transitive Property of Congruence)
• LI ∥ ON (Converse of Corresponding Angles Theorem)

The Converse of the Corresponding Angles Theorem says that if two lines and a transversal form congruent corresponding angles, then the lines are parallel.

Alternate Interior Angles in Real Life

Look at a window with panes divided by muntins. The parallel, vertical muntins are probably transversed by a horizontal muntin. Anywhere they cross, you can find alternate interior angles.

Make a capital letter Z, like the Mark of Zorro (you'll probably have to look up that 1919 superhero). The top and bottom horizontal slashes of Zorro's sword are parallel lines, and the diagonal slash is the transversal. Zorro's big Z makes two obvious, alternate interior angles.

Lesson Summary

After navigating this lesson, you are now able to define the parts of an angle (lines, rays or line segments meeting at an endpoint and forming a vertex). You can also draw, describe and identify transversal lines, straight lines, straight angles, parallel lines, and alternate interior angles.

In addition, you can now apply the Alternate Interior Angles Theorem to find angles in parallel lines crossed by a transversal. You definitely figured out the angles on this one!

Next Lesson:

Transversal Liness & Angles

CCSS-M.G-CO.C.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

After proving that vertical angles are congruent, we turned our attention towards angles formed by parallel lines cut by a transversal.

My students come to high school geometry having experience with angle measure relationships when parallel lines are cut by a transversal. But they haven’t thought about why.

We make sense of Euclid’s 5th Postulate (wording below from Cut the Knot):

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

We use dynamic geometry software to explore Parallel Lines and Transversals:

And then traditionally, we have allowed corresponding angles congruent when parallel lines are cut by a transversal as the postulate in our deductive system. It makes sense to students that the corresponding angles are congruent. Then once we’ve allowed those, it’s not too bad to prove that alternate interior angles are congruent when parallel lines are cut by a transversal.

But we wonder whether we have to let corresponding angles in as a postulate. Can we use rigid motions to show that the corresponding angles are congruent?

One student suggested constructing the midpoint, X, of segment BE. Then we created a parallel to lines m and n through X. That didn’t get us very far in showing that the corresponding angles are congruent. (image on the top left)

Another student suggested translating line m using vector BE. So we really translated more than just line m. We really translated the upper half-plan formed by line m. We used took a picture of the top part of the diagram (line m and above) and translated it using vector BE. We can see in the picture on the right, that m maps to n and the transversal maps to itself, and so we conclude (bottom left image) that ∠CBA is congruent to ∠DEB: if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Once corresponding angles are congruent, then proving alternate interior (or exterior) angles congruent or consecutive interior (or exterior) angles supplementary when two parallel lines are cut by a transversal follows using a mix of congruent vertical angles, transitive and/or substitution, Congruent Supplements.

But can we prove that alternate interior angles are congruent when parallel lines are cut by a transversal using rigid motions?

Several students suggested we could do the same translation (translating the “top” parallel line onto the “bottom” parallel line). ∠2≅∠2’ because of the translation (and because they are corresponding), and we can say that ∠2’≅∠3 since we have already proved that vertical angles are congruent. ∠2≅∠3 using the Transitive Property of Congruence. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team suggested constructing the midpoint M of segment XY (top image). They rotated the given lines and transversal 180˚ about M (bottom image). ∠2 has been carried onto ∠3 and ∠3 has been carried onto ∠2. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team constructed the same midpoint as above with a line parallel to the given lines through that midpoint. They reflected the entire diagram about that line, which created the line in red. They used the base angles of an isosceles triangle to show that alternate interior angles are congruent.

Note 1: We are still postulating that through a point not on a line there is exactly one line parallel to the given line. This is what textbooks I’ve used in the past have called the parallel postulate. And we are postulating that the distance between parallel lines is constant.

Note 2: We haven’t actually proven that the base angles of an isosceles triangle are congruent. But students definitely know it to be true from their work in middle school. The proof is coming soon.

Note 3: Many of these same ideas will show that consecutive (or same-side) interior angles are supplementary. We can use rigid motions to make the images of two consecutive interior angles form a linear pair.

After the lesson, a colleague suggested an Illustrative Mathematics task on Congruent angles made by parallel lines and a transverse, which helped me think through the validity of the arguments that my students made. As the journey continues, I find the tasks, commentary, and solutions on IM to be my own textbook – a dynamic resource for learners young and old.