Angles in the Same Segment and Angle in a Semicircle

This lesson zooms in on two of the most useful consequences of the angle-at-centre theorem: angles in the same segment of a circle are equal, and the angle in a semicircle is a right angle. Both are short to state, beautiful to prove, and indispensable in any problem involving circles.

Definitions

A segment of a circle is the region between a chord and an arc. There are two segments for each chord: the minor segment (smaller) and the major segment (larger).

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle. The chord opposite the vertex (the side connecting the endpoints of the two chords) defines the arc the angle subtends.

Theorem: Angles in the same segment are equal

Theorem. Two inscribed angles in a circle that subtend the same arc are equal.

In symbols: if $P$ and $Q$ are two points on a circle and $A, B$ are two other points on the circle (on the same side as $P$ and $Q$ relative to chord $AB$ ), then $\angle APB = \angle AQB$ .

Proof. Each of $\angle APB$ and $\angle AQB$ equals half the central angle $\angle AOB$ (where $O$ is the centre). Same arc, same central angle, same half. Q.E.D.

This is one of the most elegant equalities in geometry: the angle subtended by a chord at any point on a specific arc is constant. As $P$ slides along the arc, the angle does not change.

Theorem: Angle in a semicircle

Theorem. The angle subtended by a diameter at any point on the circle (not on the diameter) is $90^\circ$ .

In symbols: if $AB$ is a diameter and $C$ is a point on the circle, $\angle ACB = 90^\circ$ .

Proof. The diameter $AB$ subtends a central angle of $180^\circ$ (it is a straight angle through the centre). By the angle-at-centre theorem, the inscribed angle is half: $\tfrac{1}{2} \cdot 180^\circ = 90^\circ$ . Q.E.D.

A useful corollary

Corollary. If a right angle is subtended by a chord at a point of the circle, then that chord is a diameter.

Why. The chord subtends a central angle of $2 \cdot 90^\circ = 180^\circ$ , which means the chord passes through the centre , hence it is a diameter.

This corollary is the converse of the semicircle theorem. It lets you detect a diameter by spotting a right angle.

Worked examples

Example 1. Two angles $\angle APB$ and $\angle AQB$ subtend the same arc in a circle. If $\angle APB = 35^\circ$ , find $\angle AQB$ .

Same segment $\Rightarrow$ equal: $\angle AQB = 35^\circ$ .

Example 2. $AB$ is a diameter of a circle, $C$ is on the circle. Find $\angle ACB$ .

$\angle ACB = 90^\circ$ (angle in a semicircle).

Example 3. In a circle, $AB$ is a chord. $P, Q$ are on the major arc and $R$ on the minor arc. Compare $\angle APB$ , $\angle AQB$ , and $\angle ARB$ .

$\angle APB = \angle AQB$ (same segment, major arc). $\angle ARB$ is in the minor segment and is supplementary to $\angle APB$ (since $APBR$ is a cyclic quadrilateral with $\angle P + \angle R = 180^\circ$ ).

Example 4. In a circle, a chord subtends $\angle APB = 50^\circ$ on the major arc. Find the angle on the minor arc.

By the cyclic quadrilateral property (or directly): the angle on the minor arc is $180 - 50 = 130^\circ$ .

Example 5. A triangle is inscribed in a circle such that one side is a diameter. Show that the triangle is right-angled.

The angle opposite the diameter is inscribed in a semicircle, hence is $90^\circ$ .

Try it yourself

State the "angles in the same segment" theorem.
State the "angle in a semicircle" theorem.
In a circle, $\angle APB = 40^\circ$ subtended by chord $AB$ at $P$ . Find $\angle AQB$ at $Q$ on the same arc.
$AB$ is a diameter of a circle, $C$ is on the circle. $\angle BAC = 30^\circ$ . Find $\angle BCA$ .
Prove that the angle in a semicircle is a right angle.
Prove that angles in the same segment of a circle are equal.
A triangle is inscribed in a circle so that one side is a diameter. Why is the triangle right-angled?
In a circle, $\angle APB = 30^\circ$ on the major arc. Find the angle on the minor arc.
If a chord subtends a right angle at a point of the circle, the chord must be a $\ldots$ (fill in).
Two chords of a circle have the same length. Do they subtend equal angles at the centre? Equal arcs?

Pitfalls / Insight

"Same segment" means same arc , both inscribed vertices on the same side of the chord.
A diameter subtends right angles at every point of the circle. Use this to test for diameters.
The angle in the minor segment is supplementary to the angle in the major segment. This is one half of the cyclic quadrilateral theorem.

Insight. These two theorems make a circle's geometry almost telepathic: knowing one inscribed angle gives you many others "for free". Spot a diameter , the opposite inscribed angle is right. Spot an arc , every angle inscribed in its complementary arc has the same measure.