326. Angles in the Same Segment · equal angles, same chord, same arc

Try to break it

Drag C and D anywhere in the same segment. ∠ACB and ∠ADB always read the same value — each one is half the central angle ∠AOB, so they're forced equal. Pull C across the chord into the other segment and the angle jumps to the supplementary value (180° − original).

How you build it

Draw a chord with two angles in the same segment.

• Draw a circle — click the centre O, then a point on the rim to set the radius.
• Mark point A on the circle (it snaps onto the rim).
• Mark point B elsewhere on the circle. AB will be the chord.
• Join A to B to draw the chord AB.
• Mark point C on the major arc — the larger arc on one side of chord AB.
• Mark point D on the SAME major arc as C (same side of AB), so both angles sit in the same segment.
• Join A to C.
• Join B to C. Now ∠ACB stands on chord AB from C.
• Join A to D.
• Join B to D. ∠ADB stands on the same chord AB from the same arc — so ∠ADB = ∠ACB.

The proof, step by step

Prove that angles in the same segment of a circle are equal.

• Join OA and OB to form the central angle ∠AOB.
• ∠AOB = 2∠ACB (Angle at centre is double the angle at circumference).
• ∠AOB = 2∠ADB (Same theorem applies to point D).
• Therefore, 2∠ACB = 2∠ADB, which implies ∠ACB = ∠ADB.

Worked example

In a circle with centre O, a chord AB subtends an angle of 50° at a point C on the major arc. If D is another point on the same arc, find the measure of ∠ADB.

By the theorem, angles in the same segment are equal. Since C and D lie on the same segment (major arc) subtended by chord AB, ∠ADB = ∠ACB = 50°.

• 25°
• 50° — correct
• 100°
• 130°