Angles in the same segment are equal. Any two angles on the circumference of a circle that are formed from the same two points on the circumference are equal. These angles lie in the same segment — the region bounded by a chord and the arc. Look for a bowtie shape when identifying this theorem. It works both ways: angles at P and Q subtended by the same chord are also equal.
Circle Theorem: Angles in the same segment are equal.
Any two angles on the circumference formed from the same two points on the circumference are equal.
These angles lie in the same segment of the circle — the region between a chord and the arc.
Angle PAQ = Angle PBQ — both are in the same segment (above chord PQ).
Find two points on the circumference that meet at a third point.
See if there are other pairs of lines from the same two original points that meet at a different point on the circumference.
Look out for a bowtie shape.
The theorem works 'upside down' — angles at P and Q (the chord endpoints) are also equal if subtended by the same chord.
Problem: Find θ. CE is a diameter, so triangles EAC and CED are in a semicircle → angle EAC = angle CED = 90°.
Angle ECA = 64°, angle ECD = 17° (angles in a triangle = 180°).
Angle θ (EBD) and angle ECD are both formed by lines from E and D — angles in the same segment are equal → θ = 17°.
An exam diagram may have multiple equal angles — look for as many as possible.
Find pairs of lines that start from the same two points on the circumference.
Combine with the angle in a semicircle (90°) when a diameter is present.
