C4.7 Circle theorems

Learning objectives

C4.7.1Calculate unknown angles and give explanations using the following• Calculate unknown angles and give explanations using the following geometrical properties of circles: • angle in a semicircle = 90° • angle between tangent and radius = 90°. Candidates will be expected to use the geometrical properties listed in the syllabus when giving reasons for answers.

Core circle problems create a right angle in two precise configurations: a triangle built on a diameter, or a radius meeting a tangent at its contact point.

Configuration	Angle fact	Reason to state
endpoints of a diameter joined to a third point on the circumference	the angle at the third point is $90^\circ$	angle in a semicircle
radius joined to the point where a tangent touches the circle	the angle between radius and tangent is $90^\circ$	angle between tangent and radius

First confirm the diameter or the tangent contact point, then mark the guaranteed right angle. Use ordinary angle facts—such as angles in a triangle summing to $180^\circ$ or equal radii forming an isosceles triangle—to reach the unknown. Give each geometrical reason beside the step it supports.

If $DF$ is a diameter and $E$ lies on the circle, $\angle DEF=90^\circ$ . When $\angle DFE=49^\circ$ , $\angle EDF=180^\circ-90^\circ-49^\circ=41^\circ$ .

If $OB$ is a radius and $AB$ is tangent at $B$ , then $\angle OBA=90^\circ$ . With $\angle OAB=36^\circ$ , triangle $OAB$ gives $\angle AOB=180^\circ-90^\circ-36^\circ=54^\circ$ .

A chord crossing a circle is not a tangent, and a line from the centre is useful only if it reaches the tangent's contact point. For the semicircle fact, the side must be a diameter through the centre—not merely any chord.

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