The alternate segment theorem - Higher
The angle between a A straight line that just touches a point on a curve. A tangent to a circle is perpendicular to the radius which meets the tangent. and a A straight line joining two points on a circle is a chord. is equal to the angle in the alternate segment.
Example
Calculate the missing angles \(x\), \(y\) and \(z\).
The angle in a semicircle is 90°.
\(y = 90°\)
Angles in a triangle add up to 180°.
\(z = 180^\circ - 30^\circ - 90^\circ = 60^\circ\)
Using the alternate segment theorem:
angle \(x = z\)
\(x = 60^\circ\)
Proof
Let angle CDB = \(x\).
The angle between a tangent and the The distance from the centre of a circle to its circumference. The plural of radius is radii. is 90°.
Angle BDO = \(90^\circ - x\)
Triangle DOB is an Two sides have equal lengths. Angles opposite the equal sides are equal. triangle so angle DBO is \(90^\circ - x\).
Angles in a triangle add up to 180°.
Angle DOB = \(180^\circ - \text{BDO} - \text{DBO}\)
Angle DOB = \(180^\circ - (90^\circ - x) - (90^\circ - x) = 2x\)
The angle at the centre is double the angle at the circumference.
Angle DAB = \(x\)
Therefore BDC = DAB.