Properties of quadrilaterals
Find unknown angles using properties of parallelograms, rectangles, rhombuses, trapeziums and kites.
Worked examples
Finding a missing angle in a parallelogram
Straightforward
Problem
A parallelogram has one angle of . Find the size of the adjacent angle.
1
Recall the property of adjacent angles in a parallelogram.
Adjacent angles in a parallelogram are co-interior angles and sum to .
2
Write an equation for the adjacent angle.
3
Solve for the adjacent angle.
Answer
The adjacent angle is .
Finding equal angles in a kite
Moderate
Problem
Kite has and . Angles and are equal. Find the size of .
1
State the angle sum of a quadrilateral.
2
Substitute the known angles. Let .
3
Simplify the left side.
4
Solve for .
, so
Answer
.
Using algebra to find angles in a parallelogram
Challenging
Problem
In parallelogram , and . Find and the size of .
1
Recall that adjacent angles in a parallelogram are supplementary.
2
Substitute the expressions for each angle.
3
Collect like terms.
4
Solve for .
, so
5
Find by substituting .
6
Check: . Verify the angles sum to .
✓
Answer
and .
Practise
Q1·Straightforward
A parallelogram has one angle of . Find the size of the opposite angle.
Explanation
Opposite angles in a parallelogram are equal, so the opposite angle is also .
Q2·Straightforward
A parallelogram has one angle of . Find the size of the adjacent angle.
Explanation
Adjacent angles in a parallelogram are co-interior angles and sum to . So the adjacent angle is .
Q3·Straightforward
A rectangle has one angle labeled . Find the value of .
Explanation
Since all angles in a rectangle are , we have . Solving: , so .
Q4·Straightforward
Three angles of a quadrilateral measure , , and . Find the size of the fourth angle.
Explanation
The angle sum of a quadrilateral is . So the fourth angle is .
Q5·Moderate
A rhombus has one angle of . Find the size of the adjacent angle.
Explanation
In a rhombus, adjacent angles sum to (they are co-interior angles between parallel sides). So the adjacent angle is .
Q6·Moderate
A trapezium has one pair of parallel sides. One angle between the parallel sides on the same side of a non-parallel leg measures . Find the size of the co-interior angle on the same side.
Explanation
Co-interior angles between parallel lines sum to . So the co-interior angle is .
Q7·Moderate
Kite has and . Angles and are equal. Find the size of .
Explanation
. Since , each is .
Q8·Moderate
An isosceles trapezium has base angles of each. Find the size of each of the two remaining angles.
Explanation
Each base angle and its co-interior angle sum to . So each remaining angle is .
Q9·Challenging
Parallelogram has and . Find the value of .
Explanation
Adjacent angles sum to : . Simplifying: , so and .
Q10·Challenging
In quadrilateral , , , , and . Find the value of .
Explanation
. Simplifying: , so and .
Q11·Challenging
The diagonals of a rhombus bisect its angles. One diagonal bisects an angle of the rhombus, creating two half-angles of each. Find the size of the adjacent angle of the rhombus.
Explanation
The bisected angle is . Since adjacent angles in a rhombus sum to , the adjacent angle is .
Q12·Challenging
A diagonal of a parallelogram creates two alternate interior angles. The angles are and . Find the value of .
Explanation
Alternate interior angles are equal: . Solving: , so .