Constructing 90°, 60° and 45° angles

A protractor is convenient, but no human can place one exactly on $60.000\ldots°$. A compass can. Special angles like $90°$, $60°$, $45°$ and $30°$ all come out exact in principle when you build them from circles.

Idea

$90°$ angle at a given point. Suppose we need a perpendicular to a line $\ell$ at a point $O$ on it.

1. With centre $O$ and any radius, cut $\ell$ at $X$ on the left and $Y$ on the right. Now $O$ is the midpoint of $XY$.
2. Construct the perpendicular bisector of $XY$ (from the previous topic). Since $O$ is already on it, you only need one pair of arcs (above) , open the compass a bit larger and cut equal arcs from $X$ and from $Y$ meeting at $A$.
3. Join $OA$. Then $\angle AOX = 90°$.

This works because $A$ lies on the perpendicular bisector of $XY$, which we know is perpendicular to $XY$ at the midpoint $O$.

$60°$ angle at a point. This one is even simpler. A $60°$ angle is the angle of an equilateral triangle.

1. Draw a ray $AX$ from the vertex $A$.
2. With centre $A$ and any radius $r$, cut the ray at a point $B$.
3. With centre $B$ and the same radius $r$, cut an arc that meets the first arc at $C$.
4. Join $AC$. Then $\angle CAB = 60°$.

Why? Triangle $ABC$ has $AB = BC = CA = r$, so it is equilateral. All its angles are $60°$.

$45°$ angle. Build a $90°$ angle, then bisect it (next subtopic) to halve $90°$ into two $45°$ angles. A $30°$ angle comes from bisecting a $60°$ angle.

Other angles from these. Combining $90°$, $60°$, $45°$ and their bisections gives many more exact angles: $15°$ ($60°/4$), $75°$ ($30° + 45°$), $120°$ ($180° - 60°$), $150°$ ($90° + 60°$), and so on. Not every angle can be constructed with compass and ruler , a famous result says $20°$ cannot , but a surprisingly rich list can.

Worked examples

Example 1. Construct a $90°$ angle at point $P$ on a line $\ell$.

Mark $X$ and $Y$ on $\ell$ on opposite sides of $P$, both $3$ cm away. Open compass to $4$ cm; from $X$ cut an arc above $\ell$; from $Y$ cut an arc above $\ell$ with the same radius. The arcs meet at $A$. Join $PA$. $\angle APX = 90°$.

Example 2. Construct a $60°$ angle at $O$.

Draw ray $OX$. With centre $O$ and radius $4$ cm, cut $OX$ at $B$. With centre $B$ and the same $4$ cm, cut an arc that crosses the first arc at $C$. Join $OC$. $\angle COX = 60°$.

Example 3. Construct a $120°$ angle.

A $120°$ angle is the supplement of $60°$. Draw a line. At a point $O$ on the line, build a $60°$ angle on the right. The angle on the left of the ray is $180° - 60° = 120°$.

Example 4. Construct an isosceles right triangle with legs $5$ cm.

Draw $AB = 5$ cm. At $A$, construct a $90°$ angle (as in Example 1). On the perpendicular, mark $C$ such that $AC = 5$ cm. Join $BC$. Triangle $ABC$ has $\angle A = 90°$ and $AB = AC = 5$ cm , done.

Try it yourself

1. Construct a $90°$ angle at a point on a line.
2. Construct a $60°$ angle at the vertex of a ray.
3. Combine constructions to make a $120°$ angle.
4. Construct an equilateral triangle of side $4$ cm. Why does the $60°$-construction automatically give you the equilateral triangle?
5. Construct a right-angled triangle with legs $3$ cm and $4$ cm. Measure the hypotenuse. Should be close to $5$ cm , why?
6. Without using a protractor, build a square of side $5$ cm. (Hint: $90°$ angles at two adjacent corners.)
7. Can you construct an angle of $7°$ using only a compass and unmarked ruler? (Hint: $7°$ is not constructible.)
8. Explain in your own words why the $60°$ construction works.

Activity

Construct a regular hexagonal star by drawing two overlapping equilateral triangles of the same size (one pointing up, one down). You will have used the $60°$ construction six times. Colour the petals!