Geometry & measures · Angles & polygons
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Angles in parallel lines
Why corresponding angles are equal, why alternate angles are equal, and why co-interior angles add up to 180° — plus the exact naming examiners accept, and the add-a-parallel-line trick for awkward figures.
Geometry & measures · Angles & polygons
Angles in parallel lines
Why corresponding angles are equal, why alternate angles are equal, and why co-interior angles add up to 180° — plus the exact naming examiners accept, and the add-a-parallel-line trick for awkward figures.
Why it works
Parallel lines point in exactly the same direction. So when one straight line (a transversal) crosses two parallel lines, it meets each of them at the same slant. Slide the whole crossing from the first parallel line down to the second — nothing about the angles changes. The angle in the same position at each crossing is the same: corresponding angles are equal.Alternate angles are equal — the pair tucked between the parallels on opposite sides of the transversal. You can build this from what you already know: the alternate angle corresponds to one at the other crossing, and that one is vertically opposite the angle you started with. Equal, then equal again — so alternate angles match.
Co-interior angles add up to 180° — the pair between the parallels on the same side of the transversal. The corresponding angle to one of them sits on a straight line with the other, so together they make . They are not equal (unless both happen to be ) — supplementary, not matching.
The names are the reasons — and the exam wants the real names. Write "alternate angles are equal", "corresponding angles are equal", "co-interior angles add up to 180°". Letter nicknames — "Z angles", "F angles", "C angles" — score nothing on their own; examiners' reports say so every year. And check the two lines really are marked parallel (the arrowheads) before quoting any of these rules.
Awkward figures: add a parallel line. When a point sits between two parallel lines (a zigzag), draw an extra line through that point parallel to both. The awkward angle splits into two pieces, and each piece is an alternate angle with one of the given ones. Add the pieces back together at the end.