Year 9
Understand early circle theorems intuitively, explain why the angle in a semicircle is 90°, and identify angles in the same segment.
Geometry — circles and angles
Circle theorems link the geometry of a circle to angles. The angle in a semicircle theorem states that any angle inscribed in a semicircle (with the diameter as the base) is always a right angle (90°). The angles in the same segment theorem states that angles subtended by the same arc (from the same part of the circle) are equal. A cyclic quadrilateral is a quadrilateral inscribed in a circle, and its opposite angles sum to 180°.
The angle in a semicircle is always 90°
A triangle is inscribed in a semicircle with the diameter as one side. What is the angle opposite the diameter?
This works for any point on the semicircle. The diameter is always the longest side (hypotenuse), and the angle opposite it is 90°.
Angles in the same segment are equal
Two angles are inscribed in a circle and subtend the same arc. Are they equal?
The 'segment' is the region between the arc and the chord. All angles inscribed in this region (subtending the same arc) are equal.
Cyclic quadrilateral — opposite angles sum to 180°
A quadrilateral is inscribed in a circle (all four vertices on the circle). What can we say about its opposite angles?
This is a key property of quadrilaterals inscribed in circles. It means if one angle is 70°, the opposite angle must be 110°.
A semicircle has diameter AB. Point C lies on the semicircle. If angle CAB = 35°, find angle CBA.
Apply circle theorems to find unknown angles. Show which theorem you use in each step.
A triangle is inscribed in a semicircle with the diameter as its base. What is the angle at the opposite vertex?
💡 The angle in a semicircle is always 90° (a right angle).
Two angles in a circle subtend the same arc. What is their relationship?
💡 Angles in the same segment are equal.
A quadrilateral is inscribed in a circle. One angle is 65°. What is the opposite angle?
💡 Opposite angles in a cyclic quadrilateral sum to 180°: 180° - 65° = 115°.
In a semicircle with diameter PQ, point R is on the circle. What is angle PRQ?
💡 PQ is the diameter, so angle PRQ (the angle at R) is in a semicircle = 90°.
A cyclic quadrilateral has angles 70°, 80° and 120°. Find the fourth angle.
💡 The angle opposite to 90° is 180° - 90° = 90°. Check: 70° + 80° + 120° + 90° = 360°.
Watch for these when working through the lesson.
These ideas fit closely with this lesson.