Geometry & measures
Properties of Shapes: Triangles, Quadrilaterals and Symmetry
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to identify and classify different types of triangles based on their sides and angles.
- ✓By the end of this lesson students will be able to identify and classify different types of quadrilaterals based on their properties.
- ✓By the end of this lesson students will be able to calculate missing angles in triangles and quadrilaterals, providing reasons for their steps.
- ✓By the end of this lesson students will be able to identify lines of symmetry and determine the order of rotational symmetry for 2D shapes.
Key concepts
A triangle is a 2D shape with three straight sides and three interior angles. The sum of the interior angles in any triangle is always 180 degrees. Triangles can be classified by their sides or by their angles: By Sides: - Equilateral Triangle: All three sides are equal in length, and all three angles are equal (each 60 degrees). - Isosceles Triangle: Two sides are equal in length, and the two angles opposite these sides (base angles) are equal. - Scalene Triangle: All three sides are different lengths, and all three angles are different. By Angles: - Right-angled Triangle: One of its interior angles is a right angle (90 degrees). - Acute-angled Triangle: All three interior angles are acute (less than 90 degrees). - Obtuse-angled Triangle: One of its interior angles is obtuse (greater than 90 degrees but less than 180 degrees).
A quadrilateral is a 2D shape with four straight sides and four interior angles. The sum of the interior angles in any quadrilateral is always 360 degrees. Common types of quadrilaterals include: - Square: All four sides are equal in length, and all four interior angles are right angles (90 degrees). Opposite sides are parallel. - Rectangle: Opposite sides are equal in length, and all four interior angles are right angles (90 degrees). Opposite sides are parallel. - Rhombus: All four sides are equal in length. Opposite angles are equal. Opposite sides are parallel. - Parallelogram: Opposite sides are parallel and equal in length. Opposite angles are equal. - Trapezium: Has exactly one pair of parallel sides. - Kite: Has two pairs of equal-length sides that are adjacent to each other. One pair of opposite angles are equal.
A line of symmetry is an imaginary line that divides a shape into two identical halves, such that if you were to fold the shape along this line, the two halves would perfectly match. It acts like a mirror line. Some shapes have multiple lines of symmetry, while others have none.
A shape has rotational symmetry if it looks exactly the same after being rotated by less than a full turn (360 degrees) about its centre point. The 'order' of rotational symmetry is the number of times the shape looks identical during a full 360-degree rotation. A shape always has rotational symmetry of order 1 (when it returns to its original position after a 360-degree turn), but we usually only consider shapes to have rotational symmetry if the order is greater than 1.
Key facts to remember
- 1The sum of interior angles in any triangle is 180°.
- 2The sum of interior angles in any quadrilateral is 360°.
- 3An equilateral triangle has 3 equal sides, 3 equal angles (60° each), 3 lines of symmetry, and rotational symmetry of order 3.
- 4An isosceles triangle has 2 equal sides and 2 equal base angles.
- 5A square has 4 lines of symmetry and rotational symmetry of order 4.
- 6A rectangle has 2 lines of symmetry and rotational symmetry of order 2.
- 7A line of symmetry divides a shape into two mirror-image halves.
- 8The order of rotational symmetry is the number of times a shape looks identical during a 360° rotation.
Worked examples
Example 1
Calculate the size of the angle marked 'x' in the isosceles triangle below. Give a reason for each step of your working. (Imagine a triangle with two equal sides marked, and two base angles. One base angle is given as 70 degrees, and the angle at the apex is marked 'x'.)
Answer
x = 40°
Always state the geometric reasons for your angle calculations.
Example 2
Find the missing angle 'y' in the quadrilateral shown. Give a reason for your answer. (Imagine a quadrilateral with interior angles: 95°, 80°, 110°, and 'y'.)
Answer
y = 75°
Remember that the angle sum for a quadrilateral is 360°, not 180°.
Example 3
For a regular pentagon, state the number of lines of symmetry and the order of rotational symmetry.
Answer
Number of lines of symmetry = 5. Order of rotational symmetry = 5.
For any regular polygon with 'n' sides, the number of lines of symmetry is 'n', and the order of rotational symmetry is 'n'.
Common mistakes
- ✗Confusing the properties of different types of triangles, e.g., assuming an isosceles triangle has all angles equal.
- ✗Incorrectly assuming all quadrilaterals have right angles or all sides equal.
- ✗Using 180° as the sum of angles for a quadrilateral instead of 360°.
- ✗Miscounting the number of lines of symmetry or the order of rotational symmetry, especially for irregular shapes or shapes like parallelograms.
- ✗Not providing clear geometric reasons for angle calculations in exam questions.
Exam tips
- ★Always state the geometric reason for each step when calculating angles (e.g., 'angles in a triangle sum to 180°', 'base angles of an isosceles triangle are equal').
- ★Use a ruler and pencil to draw lines of symmetry accurately on diagrams, or to help visualise rotations.
- ★For rotational symmetry, imagine rotating the shape or use tracing paper to count how many times it looks the same in a full turn.
- ★Read the question carefully to identify the specific type of shape (e.g., 'isosceles triangle', 'parallelogram') as this will tell you which properties to apply.
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