Practice Maths

L20 — Classifying Triangles & Quadrilaterals

Key Terms

equilateral triangle
A triangle with 3 equal sides and 3 equal angles (each 60°).
isosceles triangle
A triangle with exactly 2 equal sides and 2 equal base angles.
scalene triangle
A triangle with no equal sides and no equal angles.
acute triangle
A triangle where all three angles are less than 90°.
right-angled triangle
A triangle containing exactly one 90° angle.
obtuse triangle
A triangle containing one angle greater than 90°.
parallelogram
A quadrilateral with two pairs of parallel and equal opposite sides.
trapezium
A quadrilateral with exactly one pair of parallel sides.
kite
A quadrilateral with two pairs of adjacent (touching) equal sides.

Classifying Triangles — by Sides

Use tick marks on diagrams to identify equal sides. Equal sides carry the same number of ticks.

Equilateral
Isosceles
Scalene
Right-angled

Classifying Triangles — by Angles

60° 60° 60° Acute
45° 90° 45° Right-angled
110° 30° 40° Obtuse

Classifying Quadrilaterals

Parallelogram
Rectangle
Rhombus
Trapezium
Kite
Double Classification: A triangle gets TWO names — one by sides and one by angles. A right-angled triangle with two equal sides is right-angled and isosceles. Always check both properties.
Quadrilateral Families: Every square is also a rectangle, a rhombus, and a parallelogram. But not every rectangle is a square — a square has the extra condition of four equal sides.

Classifying Triangles

Every triangle has three sides and three angles. Mathematicians classify triangles in two independent ways: by their sides and by their angles. These two systems are independent, so the same triangle can have two names — one from each system.

By Sides

Look at the side lengths or tick marks on the diagram:

  • Equilateral — all three sides equal. All three angles equal 60°. Three tick marks, one on each side.
  • Isosceles — exactly two sides equal. Two tick marks on the equal sides. The two base angles (opposite the equal sides) are also equal.
  • Scalene — no sides equal. No tick marks. All three angles are different.
On diagrams, equal sides are marked with the same number of ticks. One tick means one group of equal sides; a second tick means a different group. A side with no tick is a different length to all others.

By Angles

Look at the largest angle in the triangle:

  • Acute — all three angles are less than 90°. All equilateral triangles are acute.
  • Right-angled — exactly one angle is 90°. Shown on diagrams by a small square in the corner.
  • Obtuse — exactly one angle is greater than 90°.
A triangle can have at most one obtuse or right angle. All three angles must add to 180°, so if one angle is 90° or more, the other two must both be less than 90° (both acute) to keep the sum at exactly 180°.

Double Classification

Always give both names when possible:

  • A 60°/60°/60° triangle is equilateral (by sides) and acute (by angles).
  • A 90°/45°/45° triangle is isosceles (by sides) and right-angled (by angles).
  • A 110°/30°/40° triangle with no equal sides is scalene (by sides) and obtuse (by angles).

Classifying Quadrilaterals

Quadrilaterals are four-sided polygons. The key property to look for is parallel sides — sides that never meet, even if extended in both directions.

  • Parallelogram — two pairs of parallel sides. Opposite sides are equal. Opposite angles are equal.
  • Rectangle — a parallelogram with four right angles.
  • Rhombus — a parallelogram with four equal sides.
  • Square — a parallelogram with four equal sides and four right angles. A square is both a rectangle and a rhombus.
  • Trapezium — exactly one pair of parallel sides. Not a parallelogram.
  • Kite — two pairs of adjacent (touching) equal sides. Not a parallelogram.
On diagrams, parallel sides are marked with matching arrows (››). A small square in a corner indicates a right angle (90°). Tick marks show equal side lengths.

The Quadrilateral Family

Think of quadrilaterals as a family hierarchy. Parallelograms are a broad family; rectangles, rhombuses, and squares are all special parallelograms that inherit parallelogram properties and add extra conditions. Trapeziums and kites are separate families — they are not parallelograms.

A square is the most specific shape — it belongs to the parallelogram, rectangle, and rhombus families all at once. This means any fact that is true for rectangles is also true for squares, and any fact true for rhombuses is also true for squares.

  1. Identify Each Triangle — By Sides

    Name the type of triangle shown in each diagram. Use the tick marks on the sides to help you.

    (a)
    (b)
    (c)
    (d)
    1. Name the triangle in diagram (a). How do the tick marks confirm your answer?
    2. Name the triangle in diagram (b). What does the small square in the corner tell you?
    3. Name the triangle in diagram (c). Which sides are equal?
    4. Name the triangle in diagram (d). How does the absence of tick marks tell you the type?
    5. How many tick marks does an isosceles triangle have, and where are they placed?

  2. Classify Triangles — By Angles

    Classify each triangle as acute, right-angled, or obtuse. Use the labelled angles.

    60° 60° 60° (a)
    45° 90° 45° (b)
    110° 30° 40° (c)
    40° 70° 70° (d)
    1. Classify triangle (a). Which property of all three angles determines this classification?
    2. Classify triangle (b). What special angle does it contain?
    3. Classify triangle (c). Which single angle determines the classification, and what is its size?
    4. Classify triangle (d). Are all three angles within the same range? What range is that?
    5. Can a right-angled triangle also be obtuse? Use the angle sum of a triangle to explain your answer.

  3. Name Each Quadrilateral

    Name the quadrilateral shown in each diagram. Choose from: rectangle, rhombus, trapezium, kite, parallelogram.

    (a)
    (b)
    (c)
    (d)
    (e)
    1. Name the shape in diagram (a). What do the double arrow marks on its sides tell you?
    2. Name the shape in diagram (b). How many right-angle markers are shown, and what does this mean?
    3. Name the shape in diagram (c). How do the tick marks show it is different from a rectangle?
    4. Name the shape in diagram (d). How many pairs of parallel sides does it have?
    5. Name the shape in diagram (e). Describe which sides are equal and why it is not a parallelogram.

  4. True or False?

    State whether each claim is true or false. If false, rewrite it so it is correct.

    1. Every square is a rectangle.
    2. Every rectangle is a square.
    3. A rhombus always has four right angles.
    4. Every square is also a rhombus.
    5. A trapezium has two pairs of parallel sides.

  5. Isosceles Triangle Investigation

    An isosceles triangle has a top angle of 40° (the angle between the two equal sides).

    40° ? ?
    1. Calculate the two base angles of this triangle. Show your working.
    2. Classify this triangle by its angles.
    3. Tran claims: “An isosceles triangle is always acute.” Is he correct? Give a counterexample with specific angle sizes to support your answer, or explain why he is right.
    4. A different isosceles triangle has base angles of 60° each. Calculate the top angle. What special name does this triangle also have, classified by its sides?

  6. The Quadrilateral Family

    Use these four shapes: Rectangle (R), Square (S), Rhombus (Rh), Parallelogram (P).

    1. Which shapes from the list are also parallelograms?
    2. Which shapes from the list have four equal sides?
    3. Which shapes from the list have four right angles?
    4. Maya says: “A square is both a rhombus and a rectangle at the same time.” Is she correct? Justify your answer using the properties of each shape.

  7. Double Classification

    For each triangle below, give TWO classifications: one by sides and one by angles.

    60° 60° 60° (a)
    45° 90° 45° (b)
    110° 30° 40° (c)
    40° 70° 70° (d)
    1. Classify triangle (a) by sides and by angles.
    2. Classify triangle (b) by sides and by angles.
    3. Classify triangle (c) by sides and by angles. (Hint: use the angle labels to decide the sides classification.)
    4. Classify triangle (d) by sides and by angles.

  8. Name the Shape From Its Properties

    Name the quadrilateral that fits each description. There may be more than one correct answer — list all that apply.

    1. A quadrilateral with four equal sides but no right angles.
    2. A quadrilateral with exactly one pair of parallel sides.
    3. A quadrilateral with two pairs of adjacent equal sides, where opposite sides are NOT all equal.
    4. A quadrilateral with four equal sides AND four right angles.

  9. Architecture and Shapes

    An architect is designing a roof truss (a structural frame). Four sections of the truss are described below.

    • Section A: A triangle with no equal sides and one angle of exactly 90°.
    • Section B: An isosceles triangle with a top angle of 100°.
    • Section C: A quadrilateral with exactly one pair of parallel sides.
    1. Give Section A’s classification both by sides and by angles.
    2. Calculate Section B’s two base angles. Then give its classification by angles.
    3. Name the shape of Section C. Explain what property distinguishes it from a parallelogram.

  10. Shape Sorting Challenge

    Four shapes are used to tile a decorative wall panel.

    Shape 1
    Shape 2
    Shape 3
    Shape 4
    1. List all shapes that have at least one pair of parallel sides. Name the shape type for each.
    2. List all shapes that have at least two equal sides or edges.
    3. Shape 3 belongs to a special family of quadrilaterals. Name that family and explain, using its properties, why Shape 3 is a member.
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