Angle bisection and copying

Two of the most useful tricks in the compass-and-ruler toolkit are bisecting an angle (cutting it in half) and copying an angle (making an exact duplicate somewhere else). Both rely on the same idea: a clever pair of arcs creates congruent triangles.

Idea

Angle bisection. Given any angle $\angle XOY$, we want a ray $OC$ that splits it into two equal parts: $\angle XOC = \angle COY$.

Construction.

1. With centre $O$ and any radius, cut both arms of the angle. Call the cut-points $A$ (on $OX$) and $B$ (on $OY$). So $OA = OB$.
2. With centre $A$ and a sufficiently large radius, draw an arc inside the angle. With centre $B$ and the same radius, draw another arc that meets the first at $C$.
3. Join $OC$. The ray $OC$ is the bisector of $\angle XOY$.

Why? In triangles $OAC$ and $OBC$:

• $OA = OB$ (same first radius),
• $AC = BC$ (same second radius),
• $OC$ is common.

So $\triangle OAC \cong \triangle OBC$ by SSS. Corresponding angles are equal: $\angle AOC = \angle BOC$. Done.

This lets us construct $45°$ (bisect $90°$), $30°$ (bisect $60°$), $15°$ (bisect $30°$), and many more.

Copying an angle. Given $\angle A$ and a ray $XZ$, we want a new angle at $X$ equal to $\angle A$.

Construction.

1. From vertex $A$, draw an arc with any radius cutting both arms at $B$ and $C$.
2. From $X$ with the same radius, draw an arc cutting $XZ$ at $Z$.
3. Open the compass to the distance $BC$.
4. From $Z$, with this distance, cut the second arc at $Y$.
5. Join $XY$. Then $\angle YXZ = \angle BAC$.

Why? Triangles $ABC$ and $XZY$ are congruent by SSS:

• $AB = XZ$ (first radius),
• $AC = XY$ (first radius),
• $BC = ZY$ (transferred length).

So $\angle A = \angle X$. The angle has been faithfully copied.

Where this matters. Copying an angle is the key to drawing parallel lines with compass + ruler (next subtopic) and to tiling the plane with repeated shapes , you make exact copies of the angle of one tile to fit the next.

Worked examples

Example 1. Bisect an angle of $80°$.

Draw $\angle XOY = 80°$ (use a protractor for this practice only , the bisection itself uses no measurement). Cut both arms at $A, B$ with one radius. From $A$ and $B$, cut equal arcs inside the angle that meet at $C$. Join $OC$. Each new angle is $40°$.

Example 2. Construct a $45°$ angle.

First construct a $90°$ angle (previous topic). Then bisect it. Each half is $45°$.

Example 3. Construct an angle of $15°$.

$60° \to 30°$ (bisect) $\to 15°$ (bisect again). Two angle-bisections of a $60°$ angle.

Example 4. Copy a given angle to a new location.

Suppose $\angle A = 73°$. Draw an arc from $A$ cutting the arms at $B$ and $C$. On a new ray from $X$, draw the same-radius arc cutting at $Z$. Measure $BC$ with the compass (without changing it!) and cut that distance from $Z$ on the arc to find $Y$. Join $XY$: now $\angle YXZ = 73°$.

Try it yourself

1. Draw a $90°$ angle and bisect it. Measure the halves.
2. Construct an angle of $30°$ by bisecting $60°$.
3. Construct an angle of $22.5°$. (Hint: $90° \to 45° \to 22.5°$.)
4. Draw any angle. Copy it onto a different ray.
5. Bisect a straight angle ($180°$). What do you get?
6. Why does the bisection construction need $AC = BC$? Which congruence rule did we use?
7. Copy an angle of $35°$ three times in a row to make $105°$.
8. Show, using the same idea, how to construct a $75°$ angle. (Hint: $45° + 30°$.)

Activity

Take a square sheet of paper. Fold it along a diagonal to make a triangle. Fold again along the line that bisects the right-angle corner. Open the paper: the crease is the angle bisector of the right angle. With compass and ruler, can you construct this exact crease line on a fresh sheet?