21.2 Circle theorems — Practise with help

1. $180^\circ$.
2. $180^\circ$.

The centre angle is twice the circumference angle.

Calculate $2\times27$.

$2\times27^\circ=54^\circ$.

Recall the fixed angle in a semicircle and the same-segment rule.

Use the diameter for $x$, then use the common chord for $y$.

$x=90^\circ$ by the angle in a semicircle theorem. $y=x=90^\circ$ because the angles stand on the same chord in the same segment.

Opposite angles total $180^\circ$.

Set $x+103=180$.

$x=180-103=77^\circ$.

Use the triangle angle sum, not a circle theorem.

Subtract both equal angles from $180^\circ$.

$180-42-42=96^\circ$. The triangle angle sum applies because the three interior angles form a triangle.