Run the door the other way: parallel lines hand you equal F-angles, equal Z-angles, and U-angles to 180°.
Point 6 of 6 in this lesson: 14.5.6 Putting it together — multi-step reasoning
The last lesson used angles to prove two lines parallel. Now we turn the key the other way: assume the lines are parallel and read off the consequences. It is the very same three pairings from before — corresponding (F), alternate interior (Z), co-interior (U) — only now parallel is the given and the equal angles are the reward. Picture a two-way door: parallel in, equal angles out. Once you know one angle at a parallel crossing, you know all eight. That single fact is the engine behind almost every angle chase you will ever run.
Here is the headline. If a ∥ b, then every corresponding pair the transversal makes is equal. This is the converse of Test 1 from 14.4: there, equal corresponding angles proved the lines parallel; here, the parallel lines hand us the equal angles. Same fact, read in the opposite direction.
The little "feet" marks (›) on the two lines are how mathematicians say these lines are parallel. Once you see them, you are allowed to treat any corresponding pair as equal without measuring.
Corresponding-angle property. If two parallel lines are cut by a transversal, then corresponding angles are equal.
Set the one acute angle θ the transversal makes, and watch every one of the eight positions snap to either θ or 180 − θ. Two sizes. That is all parallel lines ever give you.
Now the Z. The two angles that sit between the lines on opposite sides of the transversal — the alternate interior pair — are also equal when a ∥ b.
You don't have to memorise this as a separate fact — it follows from Property 1 in one line. The marked Z-angle is vertical to a corresponding angle, and vertical angles are equal (from 14.1):
∠Z₁ = ∠(corresponding) (corresponding, a ∥ b)
∠(corresponding) = ∠Z₂ (vertical angles)
so ∠Z₁ = ∠Z₂. The alternate interior angles are equal.
The third pairing behaves differently — and it is the one students forget. The co-interior pair (the U: both inside, same side of the transversal) does not come out equal. Instead the two angles add to 180° — they are supplementary.
Why 180° and not equal? One co-interior angle is the θ angle; the other is a 180 − θ angle (its neighbour along the straight line). Add them: θ + (180 − θ) = 180°. The straight line does the work.
| Angle pair | Test (14.4): angles ⇒ parallel | Property (14.5): parallel ⇒ angles |
|---|---|---|
| Corresponding (F) | equal ⇒ ∥ | ∥ ⇒ equal |
| Alternate interior (Z) | equal ⇒ ∥ | ∥ ⇒ equal |
| Co-interior (U) | sum 180° ⇒ ∥ | ∥ ⇒ sum 180° |
Corresponding and alternate-interior angles are equal; co-interior angles are supplementary (sum 180°), not equal. Setting a co-interior pair equal is the single most common slip in this whole stage.
The table above lines up two families that look almost identical. The difference is entirely about which fact is the given and which is the conclusion.
Same picture, opposite arrow. The toggle below shows it: flip the direction and watch the given and the conclusion swap places.
Don't use a property when only the test is justified, or the reverse. If the problem never says the lines are parallel, you cannot assume equal angles — you would have to prove parallel first.
One more property, and it needs no transversal at all. If a line is parallel to a second, and that second is parallel to a third, then the first is parallel to the third:
If a ∥ b and b ∥ c, then a ∥ c.
Two lines that are both perpendicular to the same line are parallel to each other. (Both make a 90° angle with the shared line, so corresponding angles are equal — apply Test 1.)
This is where it all pays off. Real problems rarely hand you the angle you want directly; you chain rules — a parallel property, then a linear pair, then maybe a vertical-angle step — one reasoned line at a time, naming the rule on each. Step through the worked chain below.
When two lines are parallel and a transversal cuts them, all eight angles collapse into just two sizes:
a ∥ b are cut by a transversal, and one of the eight angles is 65°. List the sizes of all eight angles.
Only two sizes appear: 65° (four of them) and 115° (the other four, since 180 − 65 = 115). Every pair that shares a straight line sums to 180°.
a ∥ b. A co-interior pair measures x and 70°. Find x.
Co-interior angles are supplementary: x + 70 = 180, so x = 110°.
a ∥ b. An alternate-interior pair measures 3x and 60°. Find x.
Alternate-interior angles are equal: 3x = 60, so x = 20.
Is "equal corresponding angles" a test or a property? What is the given each way?
It is both, depending on direction. As a test, the given is "the corresponding angles are equal" and you conclude a ∥ b. As a property, the given is "a ∥ b" and you conclude the corresponding angles are equal. The given is whichever fact the problem states first.
Suppose a ∥ b and b ∥ c. What can you say about a and c?
a ∥ c — parallelism is transitive. The middle line b passes the relationship along.
a ∥ b. A transversal makes a 75° angle with line a. The angle ∠x at line b is the linear-pair neighbour of the corresponding angle. Find ∠x, naming each step.
Corresponding angle at b = 75° (corresponding, a ∥ b). Then ∠x = 180 − 75 = 105° (linear pair).
Six questions to lock it in. Tap the answer you think is right.
This lesson is the converse of 14.4, and the whole point is to keep the two directions straight. A test reasons angles → parallel; a property reasons parallel → angles. They share the same three angle pairs (corresponding, alternate interior, co-interior), so it is tempting to treat them as one fact — but in a proof you must know which statement is the given and which is the conclusion. The "two-way door" image and the side-by-side table are built to make that distinction explicit.
The specific misconception to watch for: blurring the converse — using a property (assuming equal angles) when only a test is justified, or reasoning as if non-parallel lines also yield equal angles. They do not. Equal corresponding/alternate angles require the parallel hypothesis; without it, all bets are off. A close second slip is treating co-interior angles as equal — they are supplementary, summing to 180°.
This is a first, gentle taste of multi-step deductive proof (the reason widget), which sets up triangle reasoning in Stage 15. Aligned to Common Core 8.G.A.5 (angle relationships when parallel lines are cut by a transversal) and HS G-CO.C.9 (prove theorems about lines and angles, including the parallel-lines/alternate-interior results).