Learning Objectives
- Classify triangles and quadrilaterals and use their angle and side properties.
- Construct triangles and simple quadrilaterals accurately using ruler and compasses.
- Calculate interior and exterior angles of regular and irregular polygons.
- Use formulas for polygon angle sums and determine the number of sides.
- Recognise line symmetry and rotational symmetry.
Key Terms
- Polygon
- A closed plane figure made from straight line segments.
- Regular polygon
- A polygon with equal sides and equal interior angles.
- Perpendicular bisector
- A line at right angles to a segment through its midpoint.
- Line symmetry
- Symmetry in which reflection in a line maps a shape onto itself.
- Rotational symmetry
- Symmetry in which a rotation smaller than 360 degrees maps a shape onto itself.
8th Edition Chapter Map
- Triangles and their properties
- Quadrilaterals and their properties
- Constructions of triangles and quadrilaterals
- Regular and irregular polygons
- Line and rotational symmetry
Triangles
Triangles may be classified by sides: equilateral, isosceles or scalene; and by angles: acute, right-angled or obtuse. The interior angles sum to 180^\circ. An exterior angle equals the sum of the two opposite interior angles. In an isosceles triangle, angles opposite equal sides are equal; the converse is also true.
Worked Example: Isosceles Triangle
The vertex angle of an isosceles triangle is 38^\circ. The two base angles are equal and total 180^\circ-38^\circ=142^\circ, so each is 71^\circ.
Quadrilaterals
The interior angles of any quadrilateral total 360^\circ. Important properties include:
| Quadrilateral | Key properties |
|---|---|
| Parallelogram | Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other. |
| Rectangle | A parallelogram with four right angles; diagonals are equal and bisect each other. |
| Rhombus | Four equal sides; opposite sides parallel; diagonals bisect at right angles. |
| Square | Four equal sides and four right angles; combines rectangle and rhombus properties. |
| Kite | Two pairs of adjacent equal sides; one diagonal bisects the other at right angles. |
| Trapezium | One pair of parallel sides. |
Accurate Constructions
Construction marks must remain visible. A perpendicular bisector is obtained by drawing equal-radius arcs from both endpoints of a segment and joining the intersections. An angle bisector is produced by an arc from the vertex followed by equal arcs from the points where the first arc meets the arms.
For an SSS triangle, draw one side accurately, then use compasses with radii equal to the other two sides; the arc intersection locates the third vertex. When constructing a quadrilateral, divide it into triangles and use sufficient given lengths or angles to fix each new vertex uniquely.
Construction Checklist
- Use a sharp pencil and accurate ruler scale.
- Use compasses for required arcs; do not measure a perpendicular bisector by eye.
- Keep arcs visible.
- Label points and state any measured result to the requested accuracy.
Polygons And Angle Sums
A polygon is a closed plane shape made of straight sides. A regular polygon has all sides and all interior angles equal. Drawing diagonals from one vertex divides an n-sided polygon into n-2 triangles.
Worked Example: Number Of Sides
Each exterior angle of a regular polygon is 24^\circ. Hence n=360/24=15. It is a regular 15-gon.
Symmetry
A line of symmetry divides a shape into mirror-image halves. The order of rotational symmetry is the number of times a shape matches itself during a full 360^\circ turn. A regular n-gon has n lines of symmetry and rotational symmetry of order n.
Examination Guidance
- Give angle reasons such as “base angles of an isosceles triangle” or “interior angles of a quadrilateral”.
- Do not erase construction arcs.
- Use exterior-angle methods for regular polygons when they are shorter.
- Distinguish a shape’s name from additional properties shown by marks.
Common Mistakes
- Assuming every trapezium is isosceles.
- Using n\times180^\circ for the polygon interior sum.
- Confusing the sum of exterior angles, always 360^\circ, with one exterior angle.
- Drawing a construction without compass arcs.
Chapter Practice
1. Find each interior angle of a regular 12-gon.
(12-2)180/12=150^\circ.
2. The interior angle of a regular polygon is 165^\circ. Find the number of sides.
Exterior angle =15^\circ, so n=360/15=24.
3. An isosceles triangle has one angle 112^\circ. Find the other angles.
The 112-degree angle must be the unique vertex angle. The others are (180-112)/2=34^\circ each.
Further Polygon And Construction Practice
4. The angles of a quadrilateral are x,2x,3x and 4x. Find them.
10x=360^\circ, so x=36^\circ. The angles are 36, 72, 108 and 144 degrees.
5. A regular polygon has 20 sides. Find one exterior angle and one interior angle.
Exterior =360/20=18^\circ. Interior =180-18=162^\circ.
6. Describe the construction of a triangle with sides 7 cm, 6 cm and 5 cm.
Draw a 7 cm base. With compass radius 6 cm, draw an arc from one endpoint. With radius 5 cm, draw an arc from the other endpoint. Join the arc intersection to both endpoints, leaving arcs visible.
