Lines, Angles and Pythagoras

Updated 13 October 2026

An angle chase starts with a few angles you know and works out the rest without measuring. Nearly every rule comes from two facts: angles on a straight line add to 180°, and a line crossing two parallel lines makes the same angles at both crossings.

What are the basic angle facts?

Angles at a point add to 360°, one full turn, and angles on a line add to 180°, a half turn. A missing angle is the right total minus the angles you know. If two angles sit on a straight line and one is 118°, the other is 62°, because 180 − 118 = 62. Around a point, angles of 90°, 150° and 45° leave 360 − 90 − 150 − 45 = 75, so the last angle is 75°. These facts hold because degrees measure turning: 360° is one full turn by definition, and a straight line is half of that turn. When several missing angles are equal, subtract first and then share. Three equal angles at a point beside one angle of 60° are each (360 − 60) ÷ 3 = 100 degrees. Before you answer, add all the angles, including the ones you found, and make sure they reach the total: 100 + 100 + 100 + 60 = 360.

Two angles are complementary when they add to 90°.

Where two straight lines cross, the angles facing each other across the crossing are vertically opposite, and they are equal. Each of them makes a straight line with the same neighbor, so both are 180° minus that neighbor.

What do parallel lines do to angles?

When a line crosses two parallel lines, the eight angles it makes come in only two sizes, and the two sizes add to 180°. Corresponding angles sit in matching positions at the two crossings and are equal. Alternate angles sit between the parallel lines on opposite sides of the crossing line, in a Z shape, and are equal. Co-interior angles sit between the parallel lines on the same side and add to 180°. If one angle is 65°, every angle in the figure is either 65° or 180 − 65 = 115 degrees. This works because parallel lines point in the same direction, so the crossing line meets both of them at the same angle. The rules only apply when the lines really are parallel, which a diagram shows with matching arrow marks; without the marks, angles that look equal prove nothing. A pair of co-interior angles should always add back to the straight line: 65 + 115 = 180.

Each crossing makes four angles, two of each size.

The letters F, Z and C help you spot the corresponding, alternate and co-interior pairs. See angles in parallel lines.

Why do a triangle's angles add to 180°?

A triangle's angles add to 180° because they can be moved to fit together along a straight line. Draw a line through the top corner, parallel to the base. The two base angles appear again at the top as alternate angles, one on each side. Together with the top angle, they now fill the straight line, so the three add to 180°. See angles in a triangle. The fact finds any missing angle: a triangle with angles of 50° and 60° has a third angle of 180 − 50 − 60 = 70 degrees. In a right-angled triangle, the other two angles must add to 90°. In an isosceles triangle the two base angles are equal, so subtract the top angle and halve what is left: with a top angle of 40°, each base angle is (180 − 40) ÷ 2 = 70 degrees. Adding the three angles confirms the answer: 40 + 70 + 70 = 180.

A polygon splits into triangles: pick one corner and draw a line from it to every corner it does not already touch. An n-sided polygon splits into n − 2 triangles, so its angles add to (n − 2) × 180°.

For a regular polygon, use the exterior angles instead. The exterior angle at a corner is the turn you make there, and walking all the way round is one full turn, so the exterior angles of any polygon add to 360°. With n sides, each exterior angle is $\frac{360^{\circ }}{n}$360°/n and each interior angle is 180° minus that. A regular pentagon: 360 ÷ 5 = 72°, so each interior angle is 180 − 72 = 108°.

What does Pythagoras' theorem say?

Pythagoras' theorem says that in a right-angled triangle, the square of the hypotenuse, the side opposite the right angle, equals the sum of the squares of the other two sides. Call the two shorter sides a and b and the hypotenuse c; then $a^2+b^2=c^2$a² + b² = c². If the shorter sides are 6 cm and 8 cm, square them and add: 36 + 64 = 100, so $c^2$c² is 100 and the hypotenuse is 10 cm, since 10 × 10 = 100. The theorem is about areas: squares drawn outward on the two shorter sides have the same total area as the square on the hypotenuse. See Pythagoras' theorem. The theorem needs a right angle. A triangle with sides 4, 5 and 6 has no right angle, and it shows: 16 + 25 = 41, which is not 36. The hypotenuse is always the longest side, so a shorter side that comes out longer than it means a step went wrong.

The proof cuts a square up two ways. Four copies of the triangle inside a square of side a + b leave a tilted hole of area $c^2$c². The same four triangles slid into two rectangles leave two holes of area $a^2$a² and $b^2$b². Same square, same triangles, so $a^2+b^2=c^2$a² + b² = c². See proving Pythagoras by dissection.

When the hypotenuse is known and a shorter side is missing, subtract instead of adding: the missing square is $c^2$c² minus the known square. The hypotenuse is 13 and one side is 5. Then 169 − 25 = 144, so the missing side is $\sqrt{144}=12$√144 = 12. If your answer is longer than the hypotenuse, you added when you should have subtracted.

The converse works in reverse: if the three sides satisfy $a^2+b^2=c^2$a² + b² = c², the angle opposite c is a right angle. Check 8, 15 and 17: 64 + 225 = 289 and $17^2=289$17² = 289, so the angle opposite the 17 is 90°. A builder squares a corner the same way with a 3-4-5 triangle.

What makes two shapes similar?

Two shapes are similar when one is an enlargement of the other. Every angle in one shape equals the matching angle in the other, and every side is multiplied by the same number, the scale factor. To find a missing length, match the sides that face equal angles, find the scale factor from one known pair, and multiply. A triangle has sides of 3 cm, 4 cm and 5 cm, and a similar triangle has 9 cm as its shortest side. The scale factor is 9 ÷ 3 = 3, so the other sides are 4 × 3 = 12 cm and 5 × 3 = 15 cm. For triangles, two pairs of equal angles are enough to prove similarity. Areas scale differently, by the square of the factor, so the bigger triangle has 9 times the area. Each pair of matching sides should give the same factor: 9 ÷ 3 = 12 ÷ 4 = 15 ÷ 5 = 3.

Similar triangles measure what you cannot reach. A 1 m stick casts a 40 cm shadow and a tree casts a 12 m shadow. The scale factor from stick to tree is 12 ÷ 0.4 = 30, so the tree is 30 m tall. See finding similar triangles in a figure.

How do you construct a bisector with compasses?

To bisect a line segment AB with compasses, open the compasses to more than half the length of AB. Draw an arc above and below the line from A, then the same two arcs from B at the same width. Join the two points where the arcs cross. That line, the perpendicular bisector, cuts AB in half at a right angle. See constructing a perpendicular bisector. It works because every point on an arc is exactly one compass width from its center. Each crossing point is therefore the same distance from A as from B, so both lie on the line of points equally far from A and B. The compass width must stay more than half of AB; if it is less, the arcs never meet and there is nothing to join. Measure to confirm: for a segment of 8 cm, each half should be 8 ÷ 2 = 4 cm, and the angle should read 90°.

To bisect an angle, strike one arc from the vertex that cuts both arms, then one arc of the same width from each cut, and rule from the vertex through the point where they meet. See constructing an angle bisector. The same moves drop a perpendicular from a point onto a line: one arc from the point cuts the line twice, and you bisect the segment between the cuts.

A locus is the set of all points that obey one rule. The points a fixed distance r from a center make a circle of radius r. The points equally far from A and B make the perpendicular bisector of AB. The points equally far from the two arms of an angle make the angle bisector. A region question is answered by shading where two loci overlap.

Where these rules stop working

On a sphere, the angles of a triangle add to more than 180°. Start at the North Pole, go down to the equator, along a quarter of it, and back up. The three angles add to 270°. The 180° rule depends on parallel lines, and a sphere has none.

Bisecting an angle with compasses and a straight edge is a standard construction. Trisecting one, cutting it into three equal parts with the same two tools, is impossible: people tried for two thousand years, and in 1837 it was proved that no such construction exists.

The mistakes worth naming

• Treating co-interior angles as equal. They add to 180°. Only corresponding and alternate angles are equal.
• Using Pythagoras' theorem on a triangle with no right angle. The theorem needs a right angle.
• Adding when you should subtract. If the answer is longer than the hypotenuse, you added.
• Rubbing out the construction arcs. They are the working, and marks are given for them.

The geometry runs from shapes, area and perimeter through this topic to congruence, circle theorems and transformations. Related guides: Graphs of Equations and Coordinate Geometry and Logic and Proof.