Lines, Angles and Pythagoras

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. Angles on a line add to 180°, a half turn. To find a missing angle, subtract the angles you know from the total.

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?

Parallel lines never meet, however far they run. Perpendicular lines cross at a right angle.

A line crossing two parallel lines makes four angles at each crossing. The two crossings are identical, so the eight angles have only two sizes, and those two sizes add to 180°.

Corresponding angles are in matching positions at the two crossings, and they are equal. Alternate angles are between the parallels on opposite sides of the crossing line, in a Z shape, and they are equal. Co-interior angles are between the parallels on the same side, and they add to 180°. The letters F, Z and C help you spot the pairs. See angles in parallel lines.

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

Draw a line through the top corner parallel to the base. The two base angles appear again at the top as alternate angles. Those two angles and the top angle now lie along a straight line, so they add to 180°. See angles in a triangle.

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?

In a right-angled triangle, the side opposite the right angle is the hypotenuse, and it is always the longest side. Call the two shorter sides a and b and the hypotenuse c. Then $a^2+b^2=c^2$a² + b² = c². The shorter sides are 3 and 4. Square them and add: 9 + 16 = 25, so the hypotenuse is $\sqrt{25}=5$√25 = 5. See Pythagoras' theorem.

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 the same multiple of the matching side. That multiple is the scale factor. To find a missing length, match the sides that face equal angles and multiply by the scale factor.

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 segment AB, strike an arc from A and an arc of the same width from B. Every point on an arc is one compass width from its center, so each crossing is as far from A as it is from B. Join the two crossings: that line cuts AB in half at a right angle. See constructing a perpendicular bisector.

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.

Learn this properly in the app

Each lesson linked above is a screen in Math Challenge with a diagram, a worked example and practice questions. The geometry lessons run from shapes, area and perimeter through this topic to congruence, circle theorems and transformations.