Quadrilaterals & Polygon Properties — Practice
The quadrilateral family is a ladder of conditions. Each rung adds one requirement and inherits everything above.
What these questions 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
Find the sum of the interior angles of a hexagon.
Worked answer
720
n = 6, so 4 × 180
= 720°
Problem 2
Find the sum of the interior angles of a pentagon.
Worked answer
540
(5-2) × 180
= 540°
Problem 3
Find the sum of the exterior angles of any convex polygon.
Worked answer
360
You turn through one full circle.
360°, whatever n is.
Problem 4
Find each exterior angle of a regular octagon.
Worked answer
45
360 ÷ 8 = 45°
Problem 5
Is every square a rectangle?
- Yes
- No
- Only if it is also a rhombus
- Only in coordinate geometry
Worked answer
Yes
A square is a parallelogram with four right angles.
That is exactly a rectangle, plus equal sides.
Problem 6
Find each interior angle of a regular decagon.
Worked answer
144
Exterior: 360 ÷ 10 = 36
Interior: 180 - 36 = 144°
Problem 7
In parallelogram ABCD, m∠ A = 70°. Find m∠ B.
Worked answer
110
180 - 70 = 110°
Opposite angles would be 70°.
Problem 8
In parallelogram ABCD, m∠ A = 70°. Find m∠ C.
Worked answer
70
∠ A and ∠ C are opposite.
m∠ C = 70°
Problem 9
Congruent diagonals that bisect each other mean the shape is a
- rectangle
- rhombus
- trapezoid
- kite
Worked answer
rectangle
Bisecting diagonals make it a parallelogram.
Congruent diagonals upgrade it to a rectangle.
Problem 10
Perpendicular diagonals that bisect each other mean the shape is a
- rhombus
- rectangle
- trapezoid
- square
Worked answer
rhombus
Bisecting makes it a parallelogram.
Perpendicular makes it a rhombus.
Problem 11
A trapezoid has bases 8 and 14. Find the length of its midsegment.
Worked answer
11
8 + 142 = 11
Problem 12
In parallelogram ABCD, m∠ A = (3x+10)° and m∠ C = (5x-30)°. Find x.
Worked answer
20
3x + 10 = 5x - 30
40 = 2x
x = 20
Problem 13
A polygon has interior angles summing to 1260°. How many sides does it have?
Worked answer
9
n - 2 = 7
n = 9
Problem 14
A regular polygon has each interior angle 156°. How many sides?
Worked answer
15
Exterior = 180 - 156 = 24
360 ÷ 24 = 15
Problem 15
A trapezoid has one base 9 and midsegment 13. Find the other base.
Worked answer
17
9 + b2 = 13
9 + b = 26
b = 17
Problem 16
A quadrilateral has diagonals that bisect each other, are congruent, and are perpendicular. Name it.
Worked answer
square
Bisecting: parallelogram.
Congruent: rectangle. Perpendicular: rhombus.
Both together: a square.