Find the value of θ in the diagram below, giving reasons for your answers.
There is a diameter here, splitting the circle into two semicircles.
Identify the two triangles in each semi circle and mark in the right angles using the angle in a semicircle theorem.
Find the other angles in the triangles using the rule angles in a triangle add up to 180°
In the diagram, notice how the angle θ is subtended from the same chord as the angle that is 17°.
The angle at B is 90°
because
it is the
angle
(at the circumference) in
a semi circle
The angle at C is
17°
because
the
angles in a triangle add up to 180°
θ = 17°
because
angles in the same segment are equal
Find the value of x, stating any angle facts and circle theorems you use.
Identify the triangle in the circle with all three vertices at the circumference.
One vertex of this triangle meets a tangent at the bottom, so look for the vertex inside the triangle opposite this point and mark that angle with 2 x + 5.
Give reasons for your working as you go.
The top left angle is 2 x + 5 because of the alternate segment theorem
This angle is also subtended by the same arc as the angle at the centre.
The angle at the centre = 2(2 x + 5) because of the circle theorem
‘the angle at the centre is twice the angle at the circumference’
Form an equation.
Expand the brackets and solve the equation.
Using the “alternate segment theorem” and that “angles at the centre are twice angles at the circumference”