# KS4. Geometry. Angles & Symmetry. Angles in Circles

A tangent to a circle is a line that just touches the circle without crossing it. If we draw a line from the point where the tangent touches the circle to the centre of the circle, this line will make a right angle with the tangent.

From a diameter of a circle, if we draw a triangle such that the third point of the triangle is on the circle, this will always be a right-angled triangle.

Use this information to answer questions 2 to 13 below, and then we will look at some more advanced properties of the circle which are part of the extended mathematics syllabus below.

Exercise: Find the size of the angles marked by letters.

(2.) 90º, (3.) 65º, (4.) 45º, (5.) 90º, (6.) e = 40º, f = 50º, (7.) g = 30º, (8.) h = 90º, i = 60º, (9.) j = 49º, (10.) 45º, (11.) l = 60º, m = 50º, (12.) n = 40º, (13.) 50º

Extended Work: Circle Theory

Circle Theorem 1: If we draw any chord in a circle, the angle that it subtends at the centre of the circle will be twice as large as the angle it subtends on the circle.

Circle Theorem 2: If we draw any chord in a circle, the angles that it subtends on the circle are all equal to each-other.

Exercise

Let’s use this information to find the size of the angles marked with letters in exercise 11 on pages 159 and 160 of the extended textbook:

Circle Theorem 3: In a cyclic quadrilateral, each pair of opposite angles sums to 180º

⍺+ɣ=180º and β+𝜹 = 180º

Circle Theorem 4: The angle in a semicircle is a right-angle (we already know this from above)

Exercise

Let’s apply this knowledge to calculate the size of the angles marked with letters in exercise 12 on pages 161 to 162 of the extended textbook:

Circle Theory 5

(Already considered in core work above):

Circle Theory 6

Exercise