21Acircleangles

= Circle Geometry Theorems =

Angles in a Circle

 * Recall **


 * Subtend **
 * We say that an angle is ** subtended ** by an arc if the arc forms the endpoints of the lines that make the angle.
 * In both diagrams below, the angle Ð ABC is subtended by the arc AC.


 * Theorem 1 **
 * The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc.
 * In the diagrams below, we have an arc AB

math . \qquad \qquad \bullet \quad \angle ACB = 2 \times \angle ADB math




 * Theorem 2 **
 * All angles subtended on the circumference by the same arc will be equal
 * In the diagrams below, we have an arc AB

math . \qquad \qquad \bullet \quad \angle ACB = \angle ADB math


 * Theorem 3 **
 * Any angle subtended by the diameter is a right angle.
 * In the diagram below, AB is the diameter (passes through the centre of the circle)

math \\ . \qquad \qquad \bullet \quad \angle ACB = 90^\circ \\. \\ . \qquad \qquad \bullet \quad \angle ABC + \angle BAC + 90^\circ = 180^\circ \qquad \big\{ \text{angles in a triangle add to } 180^\circ \big\} math
 * Note: ** Theorem 3 is a special case of Theorem 1.
 * The angle at the centre (180º) is twice the angle at the circumference (90º).


 * Theorem 4 **
 * If a radius is drawn to any point on the circumference.
 * A tangent which touches the circle at that point will be __**perpendicular**__ (at right angles) to the radius.

math \\ . \qquad \qquad \bullet \quad \angle OAB = 90^\circ \\. \\ . \qquad \qquad \bullet \quad OA \perp BC \qquad \qquad \big\{ \perp \text{ is notation for } \underline{\text{perpendicular}} \big\} math
 * Theorem 5 **
 * If two tangents meet at an external point. (Point **A** in the diagram)
 * The angle formed is __**bisected**__ (cut in half) by a straight line from the centre of the circle.
 * The two right angled triangles formed will be __**congruent**__ (same size, same shape)

math \\ . \qquad \qquad \bullet \quad \angle CAB = 2 \times \angle OAB \\. \\ . \qquad \qquad \bullet \quad \triangle OAB \equiv \triangle OAC \\. \\ . \qquad \qquad \bullet \quad \angle OAB + \angle AOB + 90^\circ = 180^\circ \qquad \big\{ \text{angles in a triangle add to } 180^\circ \big\} \\. \\ . \qquad \qquad \bullet \quad \angle BAC + \angle BOC + 90^\circ + 90^\circ = 360^\circ \qquad \big\{ \text{angles in a quadrilateral add to } 360^\circ \big\} math



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