Relative Sizes of Angles in Segments

Theorem

In a circle:

• the angle in a semicircle is right;
• the angle in a segment greater than a semicircle is acute, and
• the angle in a segment less than a semicircle is obtuse.

Further:

• the angle of a segment greater than a semicircle is obtuse;
• the angle of a segment less than a semicircle is acute.

Proof

Let $ABCD$ be a circle whose diameter is $BC$ and whose center is $E$.

Join $AB$, $AC$, $AD$, $DC$ and $AE$.

Let $BA$ be produced to $F$.

Since $BE = EA$, from Isosceles Triangles have Two Equal Angles it follows that $\angle ABE = \angle BAE$.

Since $CE = EA$, from Isosceles Triangles have Two Equal Angles it follows that $\angle ACE = \angle CAE$.

So from $\angle BAC = \angle ABE + \angle ACE = \angle ABC + \angle ACB$.

But from Sum of Angles of Triangle Equals Two Right Angles $\angle FAC = \angle ABC + \angle ACB$.

So $\angle BAC = \angle FAC$, and so from Book I Definition 10 each one is a right angle.

So the angle in the semicircle $BAC$ is a right angle.

$\Box$

From Two Angles of Triangle Less than Two Right Angles, in $\triangle ABC$, $\angle ABC + \angle BAC$ is less than two right angles.

As $\angle BAC$ is a right angle, it follows that $\angle ABC$ is less than a right angle.

It is also the angle in a segment $ABC$ greater than a semicircle.

Therefore the angle in a segment greater than a semicircle is acute.

$\Box$

We have that $ABCD$ is a cyclic quadrilateral.

From Opposite Angles of Cyclic Quadrilateral, $\angle ABC + \angle ADC$ equals two right angles.

As $\angle ABC$ is less than a right angle, it follows that $\angle ADC$ is greater than a right angle.

But $\angle ADC$ is the angle in a segment $ADC$ less than a semicircle.

Therefore the angle in a segment less than a semicircle is obtuse.

$\Box$

Since the angle contained by the straight lines $BA$ and $AC$ is a right angle, the angle contained by the arc $ABC$ and the straight line $AC$ is greater than a right angle.

So the angle of a segment greater than a semicircle is obtuse.

$\Box$

Since the angle contained by the straight lines $AC$ and $AF$ is a right angle, the angle contained by the arc $ADC$ and the straight line $AC$ is less than a right angle.

So the angle of a segment less than a semicircle is acute.

$\blacksquare$

Historical Note

This is Proposition 31 of Book III of Euclid's The Elements.