Properties of 2D Shapes
Understanding the properties of shapes is fundamental to geometry. From the equal sides of an equilateral triangle to the parallel sides of a rhombus, these features help us solve problems in architecture, engineering, and your GCSE Maths exam.
Properties of Triangles
A key property of all triangles: the three interior angles always add up to $180°$.
Equilateral Triangle
- All 3 sides are equal.
- All 3 angles are equal (60°).
- 3 lines of symmetry.
Isosceles Triangle
- 2 sides are equal.
- 2 base angles are equal.
- 1 line of symmetry.
Scalene Triangle
- No equal sides.
- No equal angles.
- No lines of symmetry.
Properties of Quadrilaterals
A key property of all quadrilaterals: the four interior angles always add up to $360°$.
Square
- All 4 sides equal.
- All 4 angles are 90°.
- Opposite sides are parallel.
- Diagonals bisect each other at 90°.
Rectangle
- Opposite sides are equal.
- All 4 angles are 90°.
- Opposite sides are parallel.
- Diagonals bisect each other.
Rhombus
- All 4 sides equal.
- Opposite angles are equal.
- Opposite sides are parallel.
- Diagonals bisect each other at 90°.
Parallelogram
- Opposite sides are equal.
- Opposite angles are equal.
- Opposite sides are parallel.
- Diagonals bisect each other.
Trapezium
- Exactly one pair of parallel sides.
Kite
- Two pairs of adjacent sides are equal.
- Diagonals intersect at 90°.
- One pair of opposite angles are equal.
Worked Example
Finding a missing angle in an isosceles triangle
Find the size of angle $x$.
- The tick marks show this is an isosceles triangle, so the two base angles are equal.
- Angles in a triangle sum to $180°$. Subtract the known angle: $180° – 40° = 140°$.
- The remaining $140°$ is shared equally between the two base angles.
- $x = 140° \div 2 = 70°$.
Answer: $x = 70°$
Tutor Insights
🤔 Common Misunderstandings
- Confusing a rhombus and a square: A square is a special type of rhombus where all angles are 90°.
- Confusing a parallelogram and a rectangle: A rectangle is a special parallelogram where all angles are 90°.
📝 Common Exam Mistakes
- Not giving reasons for your answers in geometry problems.
- Applying the wrong property (e.g., assuming the diagonals of a parallelogram meet at 90° — they don’t).
- Simple arithmetic errors when calculating missing angles.
Practice Questions
- Which quadrilateral has exactly one pair of parallel sides?
- A triangle has angles measuring $40°$ and $70°$. What is the size of the third angle, and what type of triangle is it?
- In a parallelogram, one angle is $75°$. What are the sizes of the other three angles?
Show Answers
- Answer: Trapezium.
- Working: Third angle $= 180° – (40° + 70°) = 70°$.
Answer: The third angle is $70°$. Since two angles are equal, it is an isosceles triangle. - Working: Opposite angles are equal, so another angle is $75°$. Consecutive angles add to $180°$, so the other two are $180° – 75° = 105°$.
Answer: $75°$, $105°$, $75°$, $105°$.
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