Geometry: Plane Figures and Angles
Plane Figures and Angles checks angle-chasing: applying parallel-line facts, triangle and polygon angle sums, and circle angle theorems in a clear order. Diagrams are usually not to scale, so mark only what is given or proved.
Core rules
- With parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.
- The angles of a triangle sum to 180°. An exterior angle equals the sum of the two remote interior angles.
- For an n-sided polygon, the interior angle sum is (n−2)×180°; the exterior angles sum to 360°.
- In a circle, the angle at the centre is twice the inscribed angle standing on the same arc. Angles in the same segment are equal. A tangent is perpendicular to the radius at the point of contact. Opposite angles of a cyclic quadrilateral sum to 180°.
Worked example
A, B, and C lie on a circle with centre O. The central angle ∠AOB=140°, and C lies on the major arc AB. Find ∠ACB. Both angles stand on the same minor arc AB. The inscribed angle is half the central angle on the same arc, so ∠ACB=140°/2=70°. If C were on the minor arc, the relevant central angle would be the reflex angle 360°−140°=220°, giving 110°.
Traps
- Using the wrong arc in an inscribed-angle question.
- Assuming a line is parallel, or a point is the centre, because it looks so.
- Confusing interior and exterior angle sums of polygons.
- Adding angles that are not subtended by the same arc.
- Treating a tangent as a chord.