One line across two makes eight angles — and three pairs worth naming: F, Z, and U.
Point 3 of 5 in this lesson: 14.3.3 Alternate interior angles — the Z
So far the action happened at a single crossing. Now lay a third line — a transversal — straight across two other lines. Each crossing makes four angles, so there are eight in all, and the interesting pairs now sit at different crossings. Today we give three of those pairs names: corresponding (the F), alternate interior (the Z), and co-interior (the U). One thing to hold onto from the start: these names are about position only — not yet about being equal. Whether the angles in a pair are equal is the headline of the next two lessons.
A transversal is simply a line that crosses two other lines. At each crossing you get the familiar family of four angles from 14.1 — a vertical pair, four linear pairs. So one transversal across two lines makes 4 + 4 = 8 angles. Going around, we number them ∠1, ∠2, ∠3, ∠4 at the top crossing and ∠5, ∠6, ∠7, ∠8 at the bottom.
The new and useful pairs are the ones that link the two crossings — one angle from the top, one from the bottom. To sort them, we first split the plane into two regions:
Two questions sort every pair: (1) is each angle interior or exterior? (2) are the two angles on the same side of the transversal, or on opposite sides? Keep those two questions handy — section 14.3.5 turns them into a table.
Stand at the top crossing and pick the angle in the upper-left spot. Now stand at the bottom crossing and pick its upper-left angle. Those two sit in the same position at their own crossing — that makes them a corresponding pair. There are four such pairs: upper-left with upper-left, upper-right with upper-right, and so on.
The shape that traces a corresponding pair is the letter F (sometimes backward or upside-down). Slide your finger along the transversal and then out along each line, and the path makes an F whose two horizontal strokes sit at matching corners.
Switch between the three pairs. Watch where the highlighted angles sit — and that they are not equal, because these two lines are not parallel.
Now look only inside the strip. Pick one interior angle at the top crossing and the interior angle at the bottom crossing that is on the opposite side of the transversal. Both interior, opposite sides — that is an alternate interior pair, and the path that joins them traces a Z.
The word alternate means "the angles switch sides of the transversal." There are two alternate-interior pairs: ∠3 with ∠6, and ∠4 with ∠5.
The same "opposite sides" idea, run on the exterior angles, gives alternate exterior angles (∠1 with ∠8, ∠2 with ∠7) — a big Z out in the open. They behave like the interior Z; we focus on the interior ones here.
Stay inside the strip, but this time pick the two interior angles on the same side of the transversal. Both interior, same side — that is a co-interior pair (you'll also hear same-side interior or allied angles). The path joining them traces a U (or a sideways C).
There are two co-interior pairs: ∠3 with ∠5, and ∠4 with ∠6. The "co-" is a reminder that the two angles stay together on one side, rather than alternating.
You never have to memorize the picture. Run the two-question test from 14.3.1 on the pair in front of you:
Question 1 — interior or exterior? Is each angle between the lines, or outside?
Question 2 — same side or opposite side of the transversal?
Those two answers pin the name down exactly:
| Pair | Letter | Interior / exterior | Side of transversal |
|---|---|---|---|
| Corresponding | F | one interior, one exterior | same side |
| Alternate interior | Z | both interior | opposite sides |
| Co-interior | U | both interior | same side |
So the two interior pairs differ only in the second answer: alternate interior = opposite sides, co-interior = same side. And corresponding is the only one of the three that pairs an interior angle with an exterior one.
A pair of angles is marked in amber. Decide what they are, then tap your answer. Cycle through the figures with the arrows.
A transversal crosses two lines and makes eight angles. The interior ones lie between the lines; the exterior ones lie outside. Three pairs link the two crossings:
Decide every pair with two questions: interior/exterior, then same/opposite side. And remember — these are position names. Whether the pair is equal waits for the parallel-line lessons.
Use the numbering from the figures above: ∠1–∠4 at the top crossing (∠1 upper-left, ∠2 upper-right, ∠3 lower-left, ∠4 lower-right), and ∠5–∠8 at the bottom crossing (∠5 upper-left, ∠6 upper-right, ∠7 lower-left, ∠8 lower-right).
∠2 sits in the upper-right position at the top crossing. Name its corresponding partner at the bottom crossing.
∠6 — the upper-right angle at the bottom crossing. Corresponding means same position at each crossing.
Name an alternate interior pair, and say in words why it earns that name.
∠3 and ∠6 (or ∠4 and ∠5). Both are interior (between the lines) and on opposite sides of the transversal — a Z.
Name a co-interior pair, and which letter it traces.
∠3 and ∠5 (or ∠4 and ∠6). Both interior, on the same side of the transversal — a U.
A pair of angles is described as "both between the lines, on opposite sides of the transversal." Classify it.
Alternate interior (the Z). Both interior rules out corresponding; opposite sides rules out co-interior.
True or false: corresponding angles are always equal. Explain.
False. They are equal only when the two lines are parallel. On non-parallel lines the corresponding angles are different sizes — "corresponding" describes position, not size.
Which guide-letter — F, Z, or U — goes with co-interior angles? And which two questions tell co-interior apart from alternate interior?
U. Both are interior, so question 1 (interior/exterior) gives the same answer; question 2 (side of transversal) separates them: co-interior is same side, alternate interior is opposite sides.
Six questions to lock it in. Tap the answer you think is right.
This lesson is pure vocabulary and position — it deliberately stops short of equality. Students learn to look at a transversal, split the plane into interior and exterior, and name the three linking pairs (corresponding/F, alternate interior/Z, co-interior/U) using two yes-or-no questions. Drawing the two lines not parallel is intentional: it lets the eye see that the named pairs are generally unequal, so the names can't secretly smuggle in "equal."
The misconception to head off: assuming the pairs are automatically equal. Many students arrive having heard "corresponding angles are equal" as a slogan and apply it before parallelism is established. Insist on the order of operations: first name the pair (this lesson), then, only when the lines are parallel, conclude equality (14.4–14.5). A quick check: ask "are these lines parallel?" before any equality claim.
Common Core: this supports 8.G.A.5 (angle relationships created when parallel lines are cut by a transversal — here, the naming groundwork) and looks ahead to HS G-CO.C.9 (theorems about lines and angles). The payoff comes next lesson, when these names become tests for parallelism.