Lines and Angles

Angles at a Point #

A full turn shared: 360.

Angles meeting at a point share one full turn of 360 degrees

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Swing a ray round by 150°. All the way round to the start is a full turn: 360°.

A second angle at the same point takes another 120° of the turn.

A third closes it: 150 + 120 + 90 = 360. Angles at a point make 360°.

One angle unknown? Subtract the others from 360.

Past 180° an angle has gone the bigger way round. That is called reflex.

Angles on a Line #

They always add to a half turn.

Angles on a straight line always add up to a half turn

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One angle here is 130°.

The rest of the straight line must be 50°, because together they make 180°.

Complementary Angles #

The pair that fills a right angle.

Two angles are complementary when they add to a right angle of 90 degrees

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A right angle measures 90°. Two angles that fill one are complementary.

Swing a ray into that corner. The part below it measures 35°.

The part left above it is 55°, because 35 + 55 comes to 90.

Missing one? Write the pair as a sum of 90 and take the known angle off.

Angles on a line fill 180°. A complementary pair fills half of that, 90°.

Vertically Opposite Angles #

Two lines cross into two equal pairs.

Where two straight lines cross, opposite angles are equal

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Two straight lines cross, and four angles open around the crossing.

The pair straight across the X are vertically opposite — and equal.

a and c each make 180° with the same b, so a and c must match.

Two equal pairs: a = c and b = d — here 70° twice and 110° twice.

Parallel and Perpendicular Lines #

Never meeting, or crossing at a square corner.

Parallel lines never meet however far they run; perpendicular lines cross at a right angle

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Top and bottom never meet, however far you run them. That is parallel.

Cross at a right angle — a square corner, 90° — and the lines are perpendicular.

A trapezium has one parallel pair. The slanted two lean together and would meet.

A square is both: opposite sides parallel, meeting sides perpendicular.

Drawing Parallel and Perpendicular Lines #

Repeat the step, or swap and flip it.

Repeat a line’s step for a parallel line; swap and flip the step for a perpendicular one

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This line steps 3 across and 2 up. Repeat that step anywhere: parallel lines.

Swap the counts and flip one: 3 across, 2 up becomes 2 across, 3 down — perpendicular.

Angles in Parallel Lines #

Matching corners are equal.

A line crossing two parallels makes equal angles in matching positions

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One line crossing two parallels makes eight angles — four at each crossing.

The parallels make the two crossings identical, so matching corners are equal.

Alternate angles sit on opposite sides in a Z, and they match as well.

Co-interior angles are the pair inside on one side, so they fill 180° together.

Angles in a Triangle #

The three always add to 180.

The three angles inside any triangle always add to a half turn

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Any triangle has three corner angles. Call them a, b and c.

Tear the three corners off and they fit one straight line: a + b + c = 180°.

Carry one side on past a corner. The outside angle e and c fill that line: e = 180 − c.

But a + b + c = 180 says 180 − c is a + b. So e = a + b, the two opposite corners.

An isosceles triangle has two equal sides, marked with a tick. The two base angles are equal too.

So one angle gives the other two: a top angle of 40° leaves 180 − 40 = 140°, and 140 ÷ 2 = 70° at each base corner.

Angles in a Polygon #

Split it into triangles and count.

Splitting a polygon into triangles gives the total of its inside angles

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Take a pentagon and pick one corner, then draw lines from it to the corners it misses.

Three triangles fill it, and every triangle holds 180°. So the pentagon holds 540°.

A hexagon fans into four. The two corners beside your own are always missed, so n − 2.

The Interior Angle of a Regular Polygon #

Share the total between corners that all match.

A regular polygon shares its angle total equally between corners that all match

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A pentagon fans into three triangles, and every triangle holds 180 degrees.

Regular means all five corners match, so each one takes a fifth of the 540.

A regular hexagon: 720 shared six ways puts 120 degrees at every corner.

Carry each side on past its corner. The turn you make there is the outside angle.

Walk the whole way round and the turns come to one full circle: 360 degrees.

The quicker method: share 360 between the corners, then subtract that from 180.

Constructing a Perpendicular Bisector #

Two arcs of one width; their crossings do the rest.

Two arcs of one compass width cross on the line that cuts a segment in half at a right angle

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This segment has no middle marked. A compass carries one width; it never measures.

Strike an arc from A. Every point on it sits that one width away from A.

The same width from B. Each crossing is as far from A as it is from B.

Join the two crossings. That line cuts the segment in half, at a right angle.

Turn the segment any way you like. The same two arcs, and the same result.

Constructing an Angle Bisector #

One arc across the arms, then two of one width.

Cutting both arms the same distance from the vertex makes a crossing that halves the angle

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Here is an angle and no protractor. Only the two arms are given.

One arc from the vertex cuts both arms the same distance out.

From each cut, one more width. The arcs meet where the two distances agree.

Rule from the vertex through that meeting point. The angle is now two equal halves.

Nothing in those moves used the size of the angle. A wide one halves the same way.

Constructing a Perpendicular from a Point #

Cut the line twice, then bisect what you cut.

An arc from the point cuts the line twice, and bisecting that cut drops the perpendicular

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This point sits above a line. The shortest way down meets the line at a right angle.

One arc from the point cuts the line twice, and both cuts are that width away.

Those two cuts are a segment. Bisect it with two equal arcs, exactly as before.

Rule from the point through the crossing. It meets the line at a right angle.

The line need not be level. Nothing in those moves looked at which way it ran.

Any other path is a hypotenuse: $c^2=d^2+b^2$c² = d² + b², so c > d. The perpendicular is shortest.

Constructing a Triangle #

Two arcs meet where the third corner must be.

Rule the base, then one arc from each end at that side’s length — the third corner is forced

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The sides are 8, 6 and 5 cm. Only one can be ruled straight away: the 8 cm base AB.

Set the compass to 6 cm and strike an arc from A. Corner C lies somewhere on it.

Reset to 5 cm and strike from B. The crossing is the one point at both distances.

Rule AC and BC. Three sides leave no freedom — every correct build is congruent.

Loci #

Every point that obeys one rule.

A locus is every point that obeys one rule, drawn as a single shape

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Every point exactly r from the center — collected, they make a circle of radius r.

Equally far from A and from B: the perpendicular bisector you constructed above.

Equally far from both arms of an angle: the angle bisector splits the gap.

A locus is the answer set, drawn. The constructions are how you draw it exactly.

Triangles on a Sphere Break the 180 Rule #

That rule belongs to flat paper only.

On a curved surface the angles of a triangle add to more than a straight line

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Start at the north pole, go down to the equator, along it, and back up.

Every corner of that journey is a 90° right angle, and there are 3 of them.

The 180° rule was never universal. It belongs to flat paper only.

Why an Angle Cannot Be Trisected #

Proved impossible, not merely never managed.

Some constructions are not merely difficult but provably impossible

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Halving an angle with compasses is a standard construction. Trisecting it is not.

For two thousand years people tried, and no one ever found a way.

It was proved impossible in 1837. Some lengths are simply out of reach.

Angle Chases #

One unknown, several rules deep.

Find one angle, use it to find the next — an angle chase runs one rule after another

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Start at 64°: alternate angles are equal, so the angle across the Z is also 64°.

Now use the 64° you just found: angles on a straight line add to 180°, so 116°.

The third angle inside is 180° − 38° − 74° = 68°, so x = 180° − 68° = 112°.