Quadrilaterals
Identify and classify quadrilaterals by their properties, and find missing angles using the angle sum of a quadrilateral.
Worked examples
Classifying a quadrilateral
Straightforward
Problem
A quadrilateral has two pairs of parallel sides, all four sides equal, and no right angles. Identify the quadrilateral.
1
Check how many pairs of parallel sides there are.
Two pairs of parallel sides means it belongs to the parallelogram family.
2
Check whether all four sides are equal.
All four sides are equal, so it is either a square or a rhombus.
3
Check whether there are any right angles.
There are no right angles, so it cannot be a square.
4
State the conclusion.
A quadrilateral with all sides equal, two pairs of parallel sides, and no right angles is a rhombus.
Answer
Rhombus
Finding a missing angle in a quadrilateral
Moderate
Problem
Three angles of a quadrilateral are , , and . Find the fourth angle.
1
Recall the angle sum of a quadrilateral.
The four angles of any quadrilateral add to .
2
Add the three known angles.
3
Subtract from .
Answer
Using algebra to find angles in a parallelogram
Challenging
Problem
A parallelogram has one angle of and the opposite angle of . Find and the size of all four angles.
1
Use the property that opposite angles in a parallelogram are equal.
2
Solve for .
, so
3
Find the size of the angle given.
4
Find the adjacent angles using co-interior angles.
Adjacent angles in a parallelogram are supplementary:
5
State all four angles.
The four angles are , , , .
Answer
; angles are , , ,
Practise
Q1·Straightforward
A quadrilateral has four right angles and all four sides equal. What type of quadrilateral is it?
Explanation
When a quadrilateral has all four sides equal and all four angles equal to , it is a square.
Q2·Straightforward
A quadrilateral has four right angles. Its opposite sides are equal in length, but not all four sides are equal. What type of quadrilateral is it?
Explanation
A rectangle has four right angles and two pairs of equal opposite sides. Unlike a square, not all four sides are equal.
Q3·Straightforward
A quadrilateral has exactly one pair of parallel sides. What type of quadrilateral is it?
Explanation
A trapezium has exactly one pair of parallel sides. A parallelogram has two pairs, so it does not qualify.
Q4·Straightforward
A quadrilateral has all four sides equal in length but no right angles. What type of quadrilateral is it?
Explanation
A rhombus has all four sides equal and opposite sides parallel. Because it has no right angles, it is not a square.
Q5·Moderate
Three angles of a quadrilateral are , , and . Find the fourth angle in degrees.
Explanation
Q6·Moderate
One angle of a rectangle is labelled . Find the value of .
Explanation
Each angle of a rectangle is .
Q7·Moderate
A parallelogram has one angle of . Find the size of the opposite angle in degrees.
Explanation
Opposite angles in a parallelogram are equal. So the angle opposite to is also .
Q8·Moderate
Three angles of a trapezium are , , and . Find the fourth angle in degrees.
Explanation
Q9·Challenging
A rhombus has one angle of and an adjacent angle of . Find the value of .
Explanation
Adjacent angles in a rhombus are co-interior angles between parallel sides, so they add to .
Q10·Challenging
In a parallelogram, one angle is twice the size of the adjacent angle. Find the size of the larger angle in degrees.
Explanation
Let the smaller adjacent angle be and the larger be . Adjacent angles in a parallelogram are supplementary:
The larger angle is .
Q11·Challenging
The four angles of a quadrilateral are in the ratio . Find the size of the largest angle in degrees.
Explanation
The largest angle is .
Q12·Challenging
Three of the four angles in a quadrilateral are , , and , and the fourth angle is . Find the value of .
Explanation