Only two sizes exist.
Four angles are 65° and four are 115°, and the two sizes always add to 180°. Once you know one angle, you know all eight — the only question is which of the two each one is.
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- Parallel Lines Cut by a Transversal
Parallel Lines Cut by a Transversal
Eight angles, but only two sizes. Every angle is one or the other.
Find the angle
Two parallel lines are cut by a transversal. One angle measures 65°. What is its co-interior (same-side interior) partner?
Co-interior angles are supplementary, not equal: 180 − 65 = 115°. A is the trap — most of the angle pairs here are equal, and co-interior is the one that is not. C uses 90 instead of 180, which is the complement rather than the supplement.
The four named pairs.
| Pair | Where | Relationship |
|---|---|---|
| Corresponding | same position at each crossing | equal |
| Alternate interior | between the lines, opposite sides | equal |
| Alternate exterior | outside the lines, opposite sides | equal |
| Co-interior | between the lines, same side | add to 180° |
F shape → corresponding
Z shape → alternate
C or U shape → co-interior
Three equal, one supplementary
Co-interior is the odd one out. If you find yourself answering "equal" to every pair, that is the one you have got wrong.
Solving for x.
alternate interior: (3x + 10)° and (5x − 30)°
they are equal
3x + 10 = 5x − 30
set equal
40 = 2x, x = 20
solve
both angles = 70°
substitute to check ✓
co-interior: (2x)° and (4x + 60)°
they add to 180
2x + 4x + 60 = 180
6x = 120, x = 20
40° and 140°
adding to 180 ✓
Running it backwards: proving lines parallel.
Each rule works in reverse as a test.
Corresponding angles equal
→ the lines are parallel
converse
Alternate interior equal
→ parallel
converse
Co-interior add to 180°
→ parallel
converse
Angles 70° and 100°
neither equal nor 180
NOT parallel
This is how a carpenter checks that two edges are parallel without measuring their whole length — one angle at each crossing is enough.
Without the parallel marks, none of it holds.
Check for the arrowheads first
Every rule in this lesson depends on the two lines being parallel. If the diagram has no arrowheads and the question does not say so, corresponding angles are simply two unrelated angles.
Parallel, one angle 118°
find all eight
four are 118°
vertical and alternate partners
four are 62°
180 − 118
Variations: an exterior pair, a two-step chain, x given to test.
Every case still comes down to "equal" or "add to 180".
Same-side exterior angles on parallel lines: (x + 40)° and (2x + 20)°
x + 40 + 2x + 20 = 180
outside, same side: they add to 180
3x = 120, so x = 40
solve
80° and 100°
adding to 180 ✓
Same-side exterior behaves like co-interior: they add to 180°, and are equal only if both are 90°.
An exterior angle is (3x)°. The interior angle vertical to it has an alternate interior partner of (x + 50)°.
vertical angle = (3x)°
vertical angles are equal
3x = x + 50
alternate interior: equal again
x = 25, each angle 75°
check: 25 + 50 = 75 ✓
Two "equal" links in a row still give "equal". Count the links that add to 180 instead.
Corresponding angles are (2x + 10)° and (3x − 20)°. Are the lines parallel when x = 25?
2(25) + 10 = 60
first angle
3(25) − 20 = 55
second angle
60 ≠ 55: not parallel
only x = 30 makes them equal
With x given, substitute and compare. Do not assume parallel and solve — that answers a different question.
On the exam
Expect a figure with parallel lines and one given angle, an angle pair written as expressions in x, or a pair of angles that tests whether the lines are parallel at all. The fastest route is to name the pair (corresponding, alternate, co-interior) and choose "equal" or "add to 180" before writing any algebra. The 10-second check: substitute x back — the two angles must be equal, or must add to exactly 180.
Now you try
Practice problem
Solve for x
Two parallel lines are cut by a transversal. Co-interior angles measure (3x + 20)° and (2x + 10)°. What is x?
Co-interior angles are supplementary: (3x + 20) + (2x + 10) = 180, so 5x + 30 = 180, giving 5x = 150 and x = 30. The angles are then 110° and 70°, which add to 180 ✓ B comes from setting them equal, which is the rule for the other three pairs.
Hint: co-interior means same-side interior — the one pair that adds to 180° rather than being equal.