Quadrilaterals & Polygon Properties — Challenge
The quadrilateral family is a ladder of conditions. Each rung adds one requirement and inherits everything above.
What these problems cover
These questions use 13 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Bisect — Cut exactly in half.
- Supplementary — Two angles summing to 90° / to 180°.
- Base — The number being multiplied. In 53 the base is 5.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Exterior angle — The angle formed by extending one side of a triangle.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Polygon — A closed figure made of straight segments that meet only at endpoints.
- Regular polygon — All sides equal and all angles equal.
- Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
- Rhombus — A parallelogram with four congruent sides.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Diagonal — A segment joining two non-adjacent vertices.
Problem 1
What do the interior angles of a quadrilateral add up to, in degrees?
- 180
- 270
- 360
- 540
- 720
Worked answer
360
A diagonal splits any quadrilateral into two triangles.
Each triangle contributes 180 degrees, so the total is 2· 180 = 360 degrees.
Problem 2
How many degrees is each interior angle of a regular hexagon?
- 60
- 90
- 108
- 120
- 135
Worked answer
120
The interior angles of a hexagon total (6-2)· 180 = 720 degrees.
A regular hexagon has six equal angles, so each is 7206 = 120 degrees.
The value 108 belongs to a regular pentagon.
Problem 3
What do the interior angles of an octagon add up to, in degrees?
- 720
- 900
- 1080
- 1260
- 1440
Worked answer
1080
An n-sided polygon splits into n-2 triangles, so its angles total (n-2)· 180 degrees.
For an octagon, n=8, giving (8-2)· 180 = 6· 180.
That is 1080 degrees.
Problem 4
How many sides does a heptagon have?
- 5
- 6
- 7
- 8
- 9
Worked answer
7
The prefix hepta means seven.
So a heptagon has seven sides. An octagon has eight and a hexagon six.
Problem 5
How many degrees is each exterior angle of a regular pentagon?
- 36
- 72
- 108
- 120
- 60
Worked answer
72
The exterior angles of any polygon add to 360 degrees, because walking round it turns through one full circle.
A regular pentagon has five equal exterior angles, so each is 3605 = 72 degrees.
The matching interior angle is 180-72 = 108 degrees.
Problem 6
A regular polygon has interior angles of 150 degrees. How many sides does it have?
- 8
- 10
- 12
- 15
- 20
Worked answer
12
Each exterior angle is 180-150 = 30 degrees, since interior and exterior angles sit on a straight line.
The exterior angles total 360 degrees, so the number of sides is 36030 = 12.
Working from the interior angle sum gives the same answer but takes longer.
Problem 7
What do the exterior angles of any convex polygon add up to, in degrees?
- 180
- 270
- 360
- 540
- It depends on the number of sides
Worked answer
360
Walking once round the outside of the polygon turns you through one complete revolution.
One revolution is 360 degrees, and that total is the same for every convex polygon.
The interior angle sum does depend on the number of sides, which is the contrast worth remembering.
Problem 8
How many degrees is each interior angle of a regular decagon?
- 120
- 135
- 144
- 150
- 160
Worked answer
144
Each exterior angle is 36010 = 36 degrees.
So each interior angle is 180-36 = 144 degrees.
Checking by the angle sum: (10-2)· 180 = 1440, and 144010 = 144.