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Polygons and Regions is taught in the note for Geometry. Read it below and look for the section on this topic.

Geometry

15 min read

Points, lines and angles

A point marks a position. A line goes on forever in both directions. A line segment has two end points. A ray starts at a point and goes on forever one way. An angle is the turn between two rays that share a starting point, called the vertex. We measure angles in degrees (°).

As the ray turns, the angle grows from zero to a straight line.
Types of angle
NameSize
Acutebetween 0° and 90°
Rightexactly 90°
Obtusebetween 90° and 180°
Straightexactly 180°
Reflexbetween 180° and 360°
Types of angleFive angles drawn from a common base line: an acute angle less than 90 degrees, a right angle of exactly 90 degrees, an obtuse angle between 90 and 180 degrees, a straight angle of 180 degrees and a reflex angle between 180 and 360 degrees.Acuteless than 90°Rightexactly 90°Obtuse90° to 180°Straightexactly 180°Reflex180° to 360°
Try itMake an angle

Slide the angle. Two angles that add to 90° are complementary; two that add to 180° are supplementary.

Worked exampleAngles on a straight line

Two angles lie on a straight line. One is . Find the other.

  1. Angles on a straight line add to 180°.
  2. Subtract.

Answer: The other angle is 65°.

Triangles

A triangle has three sides and three angles. The three angles always add up to 180°. We name triangles by their sides: equilateral (all three sides equal), isosceles (two equal sides, and the angles opposite those sides are equal) and scalene (no sides equal). A triangle with one angle of 90° is right-angled.

Types of triangleFour triangles: an equilateral triangle with three equal sides, an isosceles triangle with two equal sides, a scalene triangle with no equal sides, and a right-angled triangle with one angle of 90 degrees.Equilateral3 equal sidesIsosceles2 equal sidesScaleneno equal sidesRight-angledone angle of 90°
Angles in a triangleA triangle with interior angles a, b and c that add up to 180 degrees. The exterior angle formed by extending one side equals the sum of the two interior angles opposite to it.abcexteriora + b + c = 180° exterior angle = a + c
Worked exampleFind the third angle

Two angles of a triangle are and . Find the third angle.

  1. The three angles add up to 180°.
  2. Add the two known angles and subtract from 180°.

Answer: The third angle is 65°. Two angles are equal, so the triangle is isosceles.

Quadrilaterals

A quadrilateral has four sides. Its four angles add up to 360°, because a diagonal splits it into two triangles. The special quadrilaterals are shown below. A square is a rectangle with all sides equal; a rhombus is a parallelogram with all sides equal.

Types of quadrilateralSix four-sided shapes: a square with four equal sides and four right angles, a rectangle with equal opposite sides and four right angles, a parallelogram with two pairs of parallel sides, a rhombus with four equal sides, a trapezium with one pair of parallel sides, and a kite with two pairs of equal adjacent sides.Square4 equal sidesRectangleopposite sides equalParallelogram2 pairs parallelRhombus4 equal sidesTrapezium1 pair parallelKite2 pairs equal

Parallel lines

Parallel lines stay the same distance apart and never meet. A line that crosses them is a transversal. It makes pairs of angles with special names: corresponding angles (in matching positions) and alternate angles (on opposite sides of the transversal, between the parallel lines) are equal, and co-interior angles add up to 180°.

Angles with parallel linesTwo parallel lines cut by a transversal, with the eight angles named a to h. Corresponding angles such as a and e are equal. Alternate angles such as c and e are equal. Co-interior angles such as d and e add up to 180 degrees. Vertically opposite angles such as a and c are equal.››abcdefghCorresponding: a = e,b = f, c = g, d = hAlternate: c = e, d = fCo-interior: d + e = 180°c + f = 180°Vertically opposite:a = c, b = d
Worked exampleAngles on parallel lines

A transversal crosses two parallel lines. One angle is . Find the angle that is alternate to it, and the angle that is co-interior with it.

  1. Alternate angles are equal, so the alternate angle is 65°.
  2. Co-interior angles add up to 180°.

Answer: The alternate angle is 65° and the co-interior angle is 115°.

Polygons

Polygon
A closed shape made of straight sides. A regular polygon has all sides and all angles equal.
The interior angles of a polygon with n sides add up to S. Split the shape into n − 2 triangles from one corner.
  • number of sides
  • sum of interior angles (degrees)
Try itAngle sum of a polygon

Change the number of sides. The dashed lines split the polygon into triangles.

Angles in a polygonA regular pentagon with an exterior angle marked at each corner by extending one side. The five exterior angles of any polygon always add up to 360 degrees. Each exterior angle of this pentagon is 72 degrees and each interior angle is 108 degrees.Exterior anglesalways add up to 360°Regular pentagon:exterior = 360° ÷ 5 = 72°interior = 180° − 72° = 108°Interior sum = (n − 2) × 180°= 3 × 180° = 540°
Check yourself An angle of 135° is…
Check yourself What is the sum of the interior angles of a hexagon (6 sides)?

Key terms

Vertex
The point where two rays or sides meet.
Complementary angles
Two angles that add to 90°.
Supplementary angles
Two angles that add to 180°.
Regular polygon
A polygon with equal sides and equal angles.
Transversal
A line that crosses two or more other lines.
Isosceles triangle
A triangle with two equal sides and two equal angles.

Summary

  • Angles are measured in degrees; learn acute, right, obtuse, straight and reflex.
  • Straight line 180°, full turn 360°, vertically opposite angles equal.
  • Interior angle sum of an n-sided polygon is (n − 2) × 180°.
  • The angles of a triangle add up to 180° and of a quadrilateral to 360°.
  • Parallel lines: corresponding and alternate angles are equal; co-interior angles add up to 180°.
Points and Lines